Global dynamics of a special class of nonlinear semelparous Leslie matrix models

ABSTRACT

This paper considers the dynamics of nonlinear semelparous Leslie matrix models. First, a class of semelparous Leslie matrix models is shown to be dynamically consistent with a certain system of Kolmogorov difference equations with cyclic symmetry. Then, the global dynamics of a special class of the latter is fully determined. Combining together, we obtain a special class of semelparous Leslie matrix models which possesses generically either a globally asymptotically stable positive equilibrium or a globally asymptotically stable cycle. The result shows that the periodic behaviour observed in periodical insects can occur as a globally stable phenomenon.

KEYWORDS:

• Semelparous Leslie matrix model
• Leslie–Gower model
• Kolmogorov difference equation with cyclic symmetry
• global asymptotic stability
• asymptotic behaviour
• periodical insect

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 39A30
• 37N25
• 92D25

1. Introduction

A periodical insect means an insect whose life cycle has a fixed length of n years (n>1) and where adults only appear every nth year. Periodical cicadas are one of the most famous examples of periodical insects. To understand the mechanism that produces the periodic behaviour in periodical insects, Bulmer [4] studied a special case of the following system of difference equations: (1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) where $t\in {\mathbb{Z}}_{+}=\{0,1,2,\dots \}$. This system is a nonlinear semelparous Leslie matrix model and describes the dynamics of an age-structured population divided into n age-classes. The variable $u_i(t)$, $1\le i\le n$, denotes the number of age-i individuals at time t. For $1\le i\le n-1$, the product of the constant $s_i$ and the function ${\sigma }_i(\mathit{u})$ represents the probability that an age-i individual survives a unit of time. The product of the constant $s_n$ and the function ${\sigma }_n(\mathit{u})$ represents the fertility of an age-n individual. Here, we set all ${\sigma }_i(\mathbf{0})=1$. Thus, system (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) assumes that only the last age-class is reproductive. That is only $u_n(t)$ represents the number of adult individuals. In this sense, system (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) describes the population dynamics of semelparous species such as cicadas and beetles.

With ${\sigma }_i(\mathit{u})=\exp (-\sum _{1\le j\le n}{a}_{ij}{u}_j)$ in system (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ), Bulmer [4] gave a sufficient condition for such a system to have a locally stable cycle and reached the conclusion that periodic behaviour results if competition is more severe between age-classes than within age-classes. To mathematically verify this claim, system (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) has been studied in many papers. For example, Cushing and Li [11] studied bifurcations that occur around the extinction equilibrium of (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) with n = 2 and classified the stability of bifurcating positive equilibria and 2-cycles. This study was extended by Cushing [9] to the case n = 3; see also [8, 10, 19]. It was found that a heteroclinic cycle connecting three periodic points of a 3-cycle can also bifurcate from the extinction equilibrium. Besides these bifurcation studies, Davydova et al. [13] examined the asymptotic behaviour of (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) mathematically and numerically for the special case n = 2 and ${\sigma }_i(u_1,u_2)=\exp (-a_i(c_1{u}_1+c_2{u}_2))$, i = 1, 2. In addition, given that the coordinate axes include every cycle associated with the periodic behaviour in periodical insects, the attractivity of the coordinate axes was studied by Mjølhus et al. [25], Kon [18], Kon and Iwasa [20] and Diekmann and Planqué [14]. All these studies only reveal the local behaviour around equilibria, cycles or the coordinate axes (but see [14] for an example of (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) where the coordinate axes attract a large set of initial conditions).

This paper aims at the global behaviour of system (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ). We first try to find a certain class of (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) that is dynamically consistent with the following system of difference equations: (2) $\begin{array}{rl}{x}_1(t+1)& =g(x_1(t),x_2(t),\dots ,x_n(t))x_1(t),\\ x_2(t+1)& =g(x_2(t),x_3(t),\dots ,x_1(t))x_2(t),\\ & ⋮\\ x_n(t+1)& =g(x_n(t),x_1(t),\dots ,x_{n-1}(t))x_n(t).\end{array}$(2) That is $x_i(t+1)=g(P^{-i+1}\mathit{x}(t))x_i(t)$ for $1\le i\le n$ with (3) $P=\left(\begin{array}{ccccc}0& 0& \cdots & 0& 1\\ 1& 0& \cdots & 0& 0\\ 0& 1& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & 1& 0\end{array}\right)\mathrm{and}\mathit{x}(t)=\left(\begin{array}{c}{x}_1(t)\\ x_2(t)\\ ⋮\\ x_n(t)\end{array}\right).$(3) Here, $x_i(t)$ and $g(P^{-i+1}\mathit{x}(t))$ are, respectively, the population size and the growth rate of species i at time t.

System (2(2) $\begin{array}{rl}{x}_1(t+1)& =g(x_1(t),x_2(t),\dots ,x_n(t))x_1(t),\\ x_2(t+1)& =g(x_2(t),x_3(t),\dots ,x_1(t))x_2(t),\\ & ⋮\\ x_n(t+1)& =g(x_n(t),x_1(t),\dots ,x_{n-1}(t))x_n(t).\end{array}$(2) ) is a special system of Kolmogorov difference equations which have been used to study the population dynamics of n interacting species. There are many works on systems of Kolmogorov difference equations. See, for instance, [2, 3, 7, 17, 23, 31]. Some specific examples of system (2(2) $\begin{array}{rl}{x}_1(t+1)& =g(x_1(t),x_2(t),\dots ,x_n(t))x_1(t),\\ x_2(t+1)& =g(x_2(t),x_3(t),\dots ,x_1(t))x_2(t),\\ & ⋮\\ x_n(t+1)& =g(x_n(t),x_1(t),\dots ,x_{n-1}(t))x_n(t).\end{array}$(2) ) with n = 2 and 3 are reported in [16, 26–29]. In Theorem 2.2, it is shown that under some conditions on system (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ), there exists a sequence of non-singular matrices $A_t$ with period n such that any solutions $\mathit{u}(t)$ of (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) and $\mathit{x}(t)$ of (2(2) $\begin{array}{rl}{x}_1(t+1)& =g(x_1(t),x_2(t),\dots ,x_n(t))x_1(t),\\ x_2(t+1)& =g(x_2(t),x_3(t),\dots ,x_1(t))x_2(t),\\ & ⋮\\ x_n(t+1)& =g(x_n(t),x_1(t),\dots ,x_{n-1}(t))x_n(t).\end{array}$(2) ) satisfy (4) $\mathit{u}(t)=A_t\mathit{x}(t)\text{ for all }t\in {\mathbb{Z}}_{+}\text{ as long as }\mathit{u}(0)=A_0\mathit{x}(0).$(4) This means systems (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) and (2(2) $\begin{array}{rl}{x}_1(t+1)& =g(x_1(t),x_2(t),\dots ,x_n(t))x_1(t),\\ x_2(t+1)& =g(x_2(t),x_3(t),\dots ,x_1(t))x_2(t),\\ & ⋮\\ x_n(t+1)& =g(x_n(t),x_1(t),\dots ,x_{n-1}(t))x_n(t).\end{array}$(2) ) are dynamically consistent.

The next step is to determine the global dynamics of (2(2) $\begin{array}{rl}{x}_1(t+1)& =g(x_1(t),x_2(t),\dots ,x_n(t))x_1(t),\\ x_2(t+1)& =g(x_2(t),x_3(t),\dots ,x_1(t))x_2(t),\\ & ⋮\\ x_n(t+1)& =g(x_n(t),x_1(t),\dots ,x_{n-1}(t))x_n(t).\end{array}$(2) ). For a general system, this is not an easy task. Inspired by the Leslie–Gower competition model [22], we are led to some simplified versions of (2(2) $\begin{array}{rl}{x}_1(t+1)& =g(x_1(t),x_2(t),\dots ,x_n(t))x_1(t),\\ x_2(t+1)& =g(x_2(t),x_3(t),\dots ,x_1(t))x_2(t),\\ & ⋮\\ x_n(t+1)& =g(x_n(t),x_1(t),\dots ,x_{n-1}(t))x_n(t).\end{array}$(2) ) like the following (5) $x_i(t+1)=x_i(t)h(x_i(t)+c\sum _{j\ne i}{x}_j(t)),1\le i\le n,$(5) where the constant c means all interspecific competition coefficients are the same and the growth rate function h is assumed to be positive, continuous, strictly decreasing on $[0,\mathrm{\infty })$ with $h(0)>1>h(\mathrm{\infty })$. Furthermore, $xh(x)$ is assumed to be strictly increasing. Using some elementary comparison methods, we obtain in Sections 3 and 4 that depending on whether c<1 or not, system (5(5) $x_i(t+1)=x_i(t)h(x_i(t)+c\sum _{j\ne i}{x}_j(t)),1\le i\le n,$(5) ) and its analogues have, generically, either a globally asymptotically stable positive equilibrium or a globally asymptotically stable set of single-species equilibria.

Via the dynamical consistency relation (4(4) $\mathit{u}(t)=A_t\mathit{x}(t)\text{ for all }t\in {\mathbb{Z}}_{+}\text{ as long as }\mathit{u}(0)=A_0\mathit{x}(0).$(4) ), we finally get in Theorem 5.1 the global dynamics of semelparous Leslie matrix models (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) in which (6) ${\sigma }_i(u_1,u_2,\dots ,u_n)=(s_1{s}_2\cdots s_n{)}^{-(1/n)}h(\frac{u_i}{d_i}+c\sum _{j\ne i}\frac{u_j}{d_j}).$(6) Because of the periodicity of $A_t$ in (4(4) $\mathit{u}(t)=A_t\mathit{x}(t)\text{ for all }t\in {\mathbb{Z}}_{+}\text{ as long as }\mathit{u}(0)=A_0\mathit{x}(0).$(4) ), we will find that the set of single-species equilibria of (5(5) $x_i(t+1)=x_i(t)h(x_i(t)+c\sum _{j\ne i}{x}_j(t)),1\le i\le n,$(5) ) corresponds to a single-class n-cycle in system (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ). Thus, we obtain that depending on whether c<1 or not, system (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) has, generically, either a globally asymptotically stable positive equilibrium or a globally asymptotically stable single-class n-cycle. In particular, the case c>1 verifies Blumer's claim that periodic behaviour observed in periodical insects can occur as a globally stable cycle. Note that in (6(6) ${\sigma }_i(u_1,u_2,\dots ,u_n)=(s_1{s}_2\cdots s_n{)}^{-(1/n)}h(\frac{u_i}{d_i}+c\sum _{j\ne i}\frac{u_j}{d_j}).$(6) ), $(d_1,d_2,\dots ,d_n)$ is a positive eigenvector of the linearized system of (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) at the origin and $(s_1{s}_2\cdots s_n{)}^{1/n}$ the corresponding eigenvalue which is greater than 1 as ${\sigma }_i(\mathbf{0})=1<h(0)$ by assumption.

This paper is organized as follows. In Section 2, we derive a condition under which (4(4) $\mathit{u}(t)=A_t\mathit{x}(t)\text{ for all }t\in {\mathbb{Z}}_{+}\text{ as long as }\mathit{u}(0)=A_0\mathit{x}(0).$(4) ) holds, thus determining a certain class of semelparous Leslie matrix models (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) which is dynamically consistent with the Kolmogorov difference equations (2(2) $\begin{array}{rl}{x}_1(t+1)& =g(x_1(t),x_2(t),\dots ,x_n(t))x_1(t),\\ x_2(t+1)& =g(x_2(t),x_3(t),\dots ,x_1(t))x_2(t),\\ & ⋮\\ x_n(t+1)& =g(x_n(t),x_1(t),\dots ,x_{n-1}(t))x_n(t).\end{array}$(2) ). In Sections 3 and 4, we consider a special class of Kolmogorov difference equations that include (5(5) $x_i(t+1)=x_i(t)h(x_i(t)+c\sum _{j\ne i}{x}_j(t)),1\le i\le n,$(5) ) and completely determine its global dynamics together with the asymptotic stability of the equilibria. Combining together the results in Sections 2–4, we obtain in Section 5 the global dynamics of those semelparous Leslie matrix models whose survival functions are given in (6(6) ${\sigma }_i(u_1,u_2,\dots ,u_n)=(s_1{s}_2\cdots s_n{)}^{-(1/n)}h(\frac{u_i}{d_i}+c\sum _{j\ne i}\frac{u_j}{d_j}).$(6) ). Finally, some conclusions are given in Section 6.

2. Dynamical consistency between systems (1) and (2)

In this section, we derive a condition under which the semelparous Leslie matrix model (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) is dynamically consistent with the Kolmogorov difference Equation (2(2) $\begin{array}{rl}{x}_1(t+1)& =g(x_1(t),x_2(t),\dots ,x_n(t))x_1(t),\\ x_2(t+1)& =g(x_2(t),x_3(t),\dots ,x_1(t))x_2(t),\\ & ⋮\\ x_n(t+1)& =g(x_n(t),x_1(t),\dots ,x_{n-1}(t))x_n(t).\end{array}$(2) ). Note that system (2(2) $\begin{array}{rl}{x}_1(t+1)& =g(x_1(t),x_2(t),\dots ,x_n(t))x_1(t),\\ x_2(t+1)& =g(x_2(t),x_3(t),\dots ,x_1(t))x_2(t),\\ & ⋮\\ x_n(t+1)& =g(x_n(t),x_1(t),\dots ,x_{n-1}(t))x_n(t).\end{array}$(2) ) has cyclic symmetry in the sense that if $\mathit{x}(t)$ is a solution of (2(2) $\begin{array}{rl}{x}_1(t+1)& =g(x_1(t),x_2(t),\dots ,x_n(t))x_1(t),\\ x_2(t+1)& =g(x_2(t),x_3(t),\dots ,x_1(t))x_2(t),\\ & ⋮\\ x_n(t+1)& =g(x_n(t),x_1(t),\dots ,x_{n-1}(t))x_n(t).\end{array}$(2) ), then so is $P\mathit{x}(t)$. In fact, we have $\begin{array}{rl}(P\mathit{x}(t+1){)}_i=x_{i-1}(t+1)& =g(x_{i-1}(t),x_i(t),\dots ,x_{i+n-2}(t))x_{i-1}(t)\\ & =g(P^{-i+1}P\mathit{x}(t))(P\mathit{x}(t){)}_i,\end{array}$ where the subscripts of $x_i$ are counted mod n and $(P\mathit{x}{)}_i$ denotes the ith component of the vector $P\mathit{x}$.

Let ${\mathbb{R}}_{+}^n=\{\mathit{x}\in {\mathbb{R}}^n:x_i\ge 0\mathrm{for}\mathrm{all}i\}$ and $\mathrm{int}{\mathbb{R}}_{+}^n=\{\mathit{x}\in {\mathbb{R}}^n:x_i>0\mathrm{for}\mathrm{all}i\}$. Assume that

(H1)	$s_i>0$ and ${\sigma }_i:{\mathbb{R}}_{+}^n\to (0,\mathrm{\infty })$ with ${\sigma }_i(\mathbf{0})=1$ for $1\le i\le n$.

It is clear that both ${\mathbb{R}}_{+}^n$ and $\mathrm{int}{\mathbb{R}}_{+}^n$ are forward invariant under (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ). If each ${\sigma }_i$ is differentiable, then the linearization of system (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) at the origin yields the linear difference equation $\mathit{u}(t+1)=U\mathit{u}(t)$, where $U=\left(\begin{array}{ccccc}0& 0& \cdots & 0& s_n\\ s_1& 0& \cdots & 0& 0\\ 0& s_2& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & s_{n-1}& 0\end{array}\right)\mathrm{and}\mathit{u}(t)=\left(\begin{array}{c}{u}_1(t)\\ u_2(t)\\ ⋮\\ u_n(t)\end{array}\right).$ Being a non-negative irreducible matrix, Perron–Frobenius Theorem ensures that U has a dominant eigenvalue ${\lambda }_0>0$ and a positive eigenvector, say, $\mathit{d}=(d_1,d_2,\dots ,d_n{)}^{\mathrm{\top }}$ associated with ${\lambda }_0$. It is straightforward to show that ${\lambda }_0={\mathcal{R}}_0^{1/n}$, where ${\mathcal{R}}_0=s_1{s}_2\cdots s_n$ is called the basic reproduction number and represents the number of offspring reproduced by an individual in its lifetime when the density effects are ignored and $\{d_i\}$ satisfy (7) $\frac{d_n}{d_1}{s}_n=\frac{d_1}{d_2}{s}_1=\cdots =\frac{d_{n-1}}{d_n}{s}_{n-1}={\mathcal{R}}_0^{1/n},$(7) which implies that once $d_1$ is fixed, $d_2,d_2,\dots ,d_n$ are uniquely determined by $d_i=\frac{s_{i-1}{s}_{i-2}\cdots s_1}{{\mathcal{R}}_0^{(i-1)/n}}{d}_1,2\le i\le n.$ Because U is the Jacobian matrix of system (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) evaluated at the origin, the origin of (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) is asymptotically stable if ${\mathcal{R}}_0<1$ and unstable if ${\mathcal{R}}_0>1$. Moreover, $U\mathit{d}={\lambda }_0\mathit{d}$ implies that the vector $(1/\sum _{i=1}^n{d}_i)\mathit{d}$ gives a stationary age-distribution for the linearized system $\mathit{u}(t+1)=U\mathit{u}(t)$. In fact, if the initial population $\mathit{u}(0)$ is proportional to $\mathit{d}$, then so is $\mathit{u}(t)$ for each $t\in {\mathbb{Z}}_{+}$. This motivates us to consider the following normalized population: (8) $y_i(t)=u_i(t)/d_i\text{ for }1\le i\le n,\text{ i.e., }D\mathit{y}(t)=\mathit{u}(t),$(8) where D is the diagonal matrix whose diagonal entries are $d_1,d_2,\dots ,d_n$. We will see below how the desired dynamical consistency is obtained via $\mathit{y}(t)$. Using (7(7) $\frac{d_n}{d_1}{s}_n=\frac{d_1}{d_2}{s}_1=\cdots =\frac{d_{n-1}}{d_n}{s}_{n-1}={\mathcal{R}}_0^{1/n},$(7) ) and (8(8) $y_i(t)=u_i(t)/d_i\text{ for }1\le i\le n,\text{ i.e., }D\mathit{y}(t)=\mathit{u}(t),$(8) ), it is easy to check that (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) becomes (9) $\begin{array}{rl}{y}_1(t+1)& ={\mathcal{R}}_0^{1/n}{\sigma }_n(d_1{y}_1(t),d_2{y}_2(t),\dots ,d_n{y}_n(t))y_n(t),\\ y_2(t+1)& ={\mathcal{R}}_0^{1/n}{\sigma }_1(d_1{y}_1(t),d_2{y}_2(t),\dots ,d_n{y}_n(t))y_1(t),\\ & ⋮\\ y_n(t+1)& ={\mathcal{R}}_0^{1/n}{\sigma }_{n-1}(d_1{y}_1(t),d_2{y}_2(t),\dots ,d_n{y}_n(t))y_{n-1}(t).\end{array}$(9) We introduce the following assumption on the survival probabilities above:

(H2)	${\mathcal{R}}_0^{1/n}{\sigma }_i(D\mathit{x})=g(P^{-i+1}\mathit{x})$ for all $\mathit{x}\in {\mathbb{R}}_{+}^n$ and $1\le i\le n$.

Here, matrices P and D are given in (3(3) $P=\left(\begin{array}{ccccc}0& 0& \cdots & 0& 1\\ 1& 0& \cdots & 0& 0\\ 0& 1& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & 1& 0\end{array}\right)\mathrm{and}\mathit{x}(t)=\left(\begin{array}{c}{x}_1(t)\\ x_2(t)\\ ⋮\\ x_n(t)\end{array}\right).$(3) ) and (8(8) $y_i(t)=u_i(t)/d_i\text{ for }1\le i\le n,\text{ i.e., }D\mathit{y}(t)=\mathit{u}(t),$(8) ), respectively. Under this assumption, (9(9) $\begin{array}{rl}{y}_1(t+1)& ={\mathcal{R}}_0^{1/n}{\sigma }_n(d_1{y}_1(t),d_2{y}_2(t),\dots ,d_n{y}_n(t))y_n(t),\\ y_2(t+1)& ={\mathcal{R}}_0^{1/n}{\sigma }_1(d_1{y}_1(t),d_2{y}_2(t),\dots ,d_n{y}_n(t))y_1(t),\\ & ⋮\\ y_n(t+1)& ={\mathcal{R}}_0^{1/n}{\sigma }_{n-1}(d_1{y}_1(t),d_2{y}_2(t),\dots ,d_n{y}_n(t))y_{n-1}(t).\end{array}$(9) ) is equivalent to (10) $y_i(t+1)=g(y_{i-1}(t),y_i(t),\dots ,y_{i-2}(t))y_{i-1}(t)\mathrm{for}1\le i\le n,$(10) which would be the same as system (2(2) $\begin{array}{rl}{x}_1(t+1)& =g(x_1(t),x_2(t),\dots ,x_n(t))x_1(t),\\ x_2(t+1)& =g(x_2(t),x_3(t),\dots ,x_1(t))x_2(t),\\ & ⋮\\ x_n(t+1)& =g(x_n(t),x_1(t),\dots ,x_{n-1}(t))x_n(t).\end{array}$(2) ) if the subscripts of all y terms on the right-hand side are shifted forward by one.

Denote by $\mathit{F}$ and $\mathit{G}$ the maps defined by (2(2) $\begin{array}{rl}{x}_1(t+1)& =g(x_1(t),x_2(t),\dots ,x_n(t))x_1(t),\\ x_2(t+1)& =g(x_2(t),x_3(t),\dots ,x_1(t))x_2(t),\\ & ⋮\\ x_n(t+1)& =g(x_n(t),x_1(t),\dots ,x_{n-1}(t))x_n(t).\end{array}$(2) ) and (9(9) $\begin{array}{rl}{y}_1(t+1)& ={\mathcal{R}}_0^{1/n}{\sigma }_n(d_1{y}_1(t),d_2{y}_2(t),\dots ,d_n{y}_n(t))y_n(t),\\ y_2(t+1)& ={\mathcal{R}}_0^{1/n}{\sigma }_1(d_1{y}_1(t),d_2{y}_2(t),\dots ,d_n{y}_n(t))y_1(t),\\ & ⋮\\ y_n(t+1)& ={\mathcal{R}}_0^{1/n}{\sigma }_{n-1}(d_1{y}_1(t),d_2{y}_2(t),\dots ,d_n{y}_n(t))y_{n-1}(t).\end{array}$(9) ), respectively. Similarly, the t-fold compositions of $\mathit{F}$ and $\mathit{G}$ with themselves are denoted by ${\mathit{F}}^t$ and ${\mathit{G}}^t$, respectively. By definition, both ${\mathit{F}}^0$ and ${\mathit{G}}^0$ mean the identity map.

Lemma 2.1

Assume that (H1) and (H2) hold. Then $\mathit{F}$ and $\mathit{G}$ are maps from ${\mathbb{R}}_{+}^n$ to itself and (11) ${\mathit{G}}^t(\mathit{x})=P^t{\mathit{F}}^t(\mathit{x})\text{ for all }\mathit{x}\in {\mathbb{R}}_{+}^n\text{ and }t\in {\mathbb{Z}}_{+}.$(11) In particular, ${\mathit{G}}^{nk}(\mathit{x})={\mathit{F}}^{nk}(\mathit{x})$ for all $\mathit{x}\in {\mathbb{R}}_{+}^n$ and $k\in {\mathbb{Z}}_{+}$.

Proof.

Clearly, $\mathit{F}$ and $\mathit{G}$ are maps from ${\mathbb{R}}_{+}^n$ to itself. Since ${\mathit{F}}^0$ and ${\mathit{G}}^0$ are the identity map, (11(11) ${\mathit{G}}^t(\mathit{x})=P^t{\mathit{F}}^t(\mathit{x})\text{ for all }\mathit{x}\in {\mathbb{R}}_{+}^n\text{ and }t\in {\mathbb{Z}}_{+}.$(11) ) holds trivially for t = 0. We show now that it holds for t = 1, so that we may use mathematical induction. The cyclic symmetry of (2(2) $\begin{array}{rl}{x}_1(t+1)& =g(x_1(t),x_2(t),\dots ,x_n(t))x_1(t),\\ x_2(t+1)& =g(x_2(t),x_3(t),\dots ,x_1(t))x_2(t),\\ & ⋮\\ x_n(t+1)& =g(x_n(t),x_1(t),\dots ,x_{n-1}(t))x_n(t).\end{array}$(2) ) implies that (12) $P\mathit{F}(\mathit{x})=\mathit{F}(P\mathit{x})\text{ for all }\mathit{x}\in {\mathbb{R}}_{+}^n.$(12) Then assumption (H2) ensures that for all $\mathit{x}\in {\mathbb{R}}_{+}^n$, (13) $\mathit{G}(\mathit{x})=\left(\begin{array}{c}{\mathcal{R}}_0^{1/n}{\sigma }_n(D\mathit{x})x_n\\ {\mathcal{R}}_0^{1/n}{\sigma }_1(D\mathit{x})x_1\\ ⋮\\ {\mathcal{R}}_0^{1/n}{\sigma }_{n-1}(D\mathit{x})x_{n-1}\end{array}\right)=\left(\begin{array}{c}g(P^{-n+1}\mathit{x})x_n\\ g(\mathit{x})x_1\\ ⋮\\ g(P^{-n+2}\mathit{x})x_{n-1}\end{array}\right)=P\mathit{F}(\mathit{x}).$(13) That verifies (11(11) ${\mathit{G}}^t(\mathit{x})=P^t{\mathit{F}}^t(\mathit{x})\text{ for all }\mathit{x}\in {\mathbb{R}}_{+}^n\text{ and }t\in {\mathbb{Z}}_{+}.$(11) ) for t = 1. Suppose ${\mathit{G}}^t(\mathit{x})=P^t{\mathit{F}}^t(\mathit{x})$ holds for some $t\ge 1$. Then ${\mathit{G}}^{t+1}(\mathit{x})=\mathit{G}({\mathit{G}}^t(\mathit{x}))=\mathit{G}(P^t{\mathit{F}}^t(\mathit{x}))=P\mathit{F}(P^t{\mathit{F}}^t(\mathit{x})),$ where (13(13) $\mathit{G}(\mathit{x})=\left(\begin{array}{c}{\mathcal{R}}_0^{1/n}{\sigma }_n(D\mathit{x})x_n\\ {\mathcal{R}}_0^{1/n}{\sigma }_1(D\mathit{x})x_1\\ ⋮\\ {\mathcal{R}}_0^{1/n}{\sigma }_{n-1}(D\mathit{x})x_{n-1}\end{array}\right)=\left(\begin{array}{c}g(P^{-n+1}\mathit{x})x_n\\ g(\mathit{x})x_1\\ ⋮\\ g(P^{-n+2}\mathit{x})x_{n-1}\end{array}\right)=P\mathit{F}(\mathit{x}).$(13) ) was used for the last equality above. Applying (12(12) $P\mathit{F}(\mathit{x})=\mathit{F}(P\mathit{x})\text{ for all }\mathit{x}\in {\mathbb{R}}_{+}^n.$(12) ) repeatedly, we obtain $P\mathit{F}(P^t{\mathit{F}}^t(\mathit{x}))=P^{t+1}{\mathit{F}}^{t+1}(\mathit{x})$ and thus ${\mathit{G}}^{t+1}(\mathit{x})=P^{t+1}{\mathit{F}}^{t+1}(\mathit{x}).$ This completes the proof of (11(11) ${\mathit{G}}^t(\mathit{x})=P^t{\mathit{F}}^t(\mathit{x})\text{ for all }\mathit{x}\in {\mathbb{R}}_{+}^n\text{ and }t\in {\mathbb{Z}}_{+}.$(11) ) by mathematical induction. The final assertion follows from the fact that $P^{nk}$ is the identity matrix for any $k\in {\mathbb{Z}}_{+}$.

The desired dynamical consistency relation (4(4) $\mathit{u}(t)=A_t\mathit{x}(t)\text{ for all }t\in {\mathbb{Z}}_{+}\text{ as long as }\mathit{u}(0)=A_0\mathit{x}(0).$(4) ) with $A_t=D{P}^t$ follows from Lemma 2.1.

Theorem 2.2

Assume (H1) and (H2) hold. Let $\mathit{u}(t)$ and $\mathit{x}(t)$ be solutions of (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) and (2(2) $\begin{array}{rl}{x}_1(t+1)& =g(x_1(t),x_2(t),\dots ,x_n(t))x_1(t),\\ x_2(t+1)& =g(x_2(t),x_3(t),\dots ,x_1(t))x_2(t),\\ & ⋮\\ x_n(t+1)& =g(x_n(t),x_1(t),\dots ,x_{n-1}(t))x_n(t).\end{array}$(2) ), respectively. Then $\mathit{u}(t)=D{P}^t\mathit{x}(t)$ for all $t\in {\mathbb{Z}}_{+}$ whenever $\mathit{u}(0)=D\mathit{x}(0)$. Here, matrices P and D are given in (3(3) $P=\left(\begin{array}{ccccc}0& 0& \cdots & 0& 1\\ 1& 0& \cdots & 0& 0\\ 0& 1& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & 1& 0\end{array}\right)\mathrm{and}\mathit{x}(t)=\left(\begin{array}{c}{x}_1(t)\\ x_2(t)\\ ⋮\\ x_n(t)\end{array}\right).$(3) ) and (8(8) $y_i(t)=u_i(t)/d_i\text{ for }1\le i\le n,\text{ i.e., }D\mathit{y}(t)=\mathit{u}(t),$(8) ), respectively.

Proof.

By (8(8) $y_i(t)=u_i(t)/d_i\text{ for }1\le i\le n,\text{ i.e., }D\mathit{y}(t)=\mathit{u}(t),$(8) ), the assumption $\mathit{u}(0)=D\mathit{x}(0)$ implies $\mathit{y}(0)=\mathit{x}(0)$. Thus, by Lemma 2.1, $G^t(\mathit{y}(0))=P^t{F}^t(\mathit{x}(0))$ holds for all $t\in {\mathbb{Z}}_{+}$. This implies that $\mathit{y}(t)=P^t\mathit{x}(t)$ for all $t\in {\mathbb{Z}}_{+}$. By (8(8) $y_i(t)=u_i(t)/d_i\text{ for }1\le i\le n,\text{ i.e., }D\mathit{y}(t)=\mathit{u}(t),$(8) ), we finally obtain that $\mathit{u}(t)=D{P}^t\mathit{x}(t)$ for all $t\in {\mathbb{Z}}_{+}$.

3. Global dynamics of Kolmogorov difference equations

As shown in [16], system (2(2) $\begin{array}{rl}{x}_1(t+1)& =g(x_1(t),x_2(t),\dots ,x_n(t))x_1(t),\\ x_2(t+1)& =g(x_2(t),x_3(t),\dots ,x_1(t))x_2(t),\\ & ⋮\\ x_n(t+1)& =g(x_n(t),x_1(t),\dots ,x_{n-1}(t))x_n(t).\end{array}$(2) ) can have a rich dynamics even under the cyclic symmetry restriction. In order to obtain some results on the global dynamics, we will study in this section the following special case of (2(2) $\begin{array}{rl}{x}_1(t+1)& =g(x_1(t),x_2(t),\dots ,x_n(t))x_1(t),\\ x_2(t+1)& =g(x_2(t),x_3(t),\dots ,x_1(t))x_2(t),\\ & ⋮\\ x_n(t+1)& =g(x_n(t),x_1(t),\dots ,x_{n-1}(t))x_n(t).\end{array}$(2) ): For $1\le i\le n$ and $t\in {\mathbb{Z}}_{+}=\{0,1,2,\dots \}$, (14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) where all $c_{\ell }\ge 0$. Equation (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ) means that the effect of the other species on the growth rate of species i is determined by their total population size $\sum _{j\ne i}{x}_j(t)$. We study system (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ) under the assumptions that for $1\le \ell \le m$,

(A1)	$h_{\ell }(x)$ are positive, continuous and strictly decreasing functions on $[0,\mathrm{\infty })$,
(A2)	there are positive constants ${\alpha }_{\ell }$ with $\sum _{\ell =1}^m{\alpha }_{\ell }=1$ such that $x^{{\alpha }_{\ell }}{h}_{\ell }(x)$ are increasing functions on $[0,\mathrm{\infty })$,
(A3)	$\prod _{\ell =1}^m{h}_{\ell }(0)>1>\underset{x\to \mathrm{\infty }}{lim}\prod _{\ell =1}^m{h}_{\ell }(x)$. In particular,

(15) $\text{there exist }L>0\text{ and }\lambda \in (0,1)\text{ such that }\prod _{\ell =1}^m{h}_{\ell }(x)<\lambda \text{ for all }x\ge L.$(15) When m = 1, (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ) is reduced to (5(5) $x_i(t+1)=x_i(t)h(x_i(t)+c\sum _{j\ne i}{x}_j(t)),1\le i\le n,$(5) ). Assumptions (A1)–(A3) above then become

(A1)′	$h(x)$ is positive, continuous and strictly decreasing on $[0,\mathrm{\infty })$,
(A2)′	$xh(x)$ is increasing on $[0,\mathrm{\infty })$,
(A3)′	$h(0)>1>\underset{x\to \mathrm{\infty }}{lim}h(x)$.

It is straightforward to show that the functions (16) $h_{\ell }(x)={({\beta }_{\ell }+\frac{a_{\ell }}{1+x})}^{{\alpha }_{\ell }},1\le \ell \le m,$(16) satisfy (A1)–(A3) if all ${\beta }_{\ell },a_{\ell }$ and ${\alpha }_{\ell }$ are positive constants with $\sum _{\ell =1}^m{\alpha }_{\ell }=1$ and $\prod _{\ell =1}^m({\beta }_{\ell }+a_{\ell }{)}^{{\alpha }_{\ell }}>1>\prod _{\ell =1}^m{{\beta }_{\ell }}^{{\alpha }_{\ell }}$. Thus, the following system is an example of (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ) satisfying (A1)–(A3): For $1\le i\le n$ and $t\in {\mathbb{Z}}_{+}=\{0,1,2,\dots \}$, (17) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{({\beta }_{\ell }+\frac{a_{\ell }}{1+x_i(t)+c_{\ell }\sum _{j\ne i}{x}_j(t)})}^{{\alpha }_{\ell }}.$(17) When m = 1, ${\beta }_1=0$, $a_1=a>1$, and $c_1=c\ge 0$, system (17(17) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{({\beta }_{\ell }+\frac{a_{\ell }}{1+x_i(t)+c_{\ell }\sum _{j\ne i}{x}_j(t)})}^{{\alpha }_{\ell }}.$(17) ) becomes (18) $x_i(t+1)=\frac{a{x}_i(t)}{1+x_i(t)+c\sum _{j\ne i}{x}_j(t)},1\le i\le n,$(18) which satisfies (A1)${}^{\mathrm{\prime }}$ –(A3)${}^{\mathrm{\prime }}$ and is a special case of the Leslie–Gower model [22]: (19) $x_i(t+1)=\frac{{\lambda }_i{x}_i(t)}{1+x_i(t)+\sum _{j\ne i}{c}_{ij}{x}_j(t)},1\le i\le n.$(19) Here, we may assume without loss of generality that the carrying capacities ${\lambda }_i-1$ are positive and strictly decreasing in i. For n = 2, Cushing et al. [7] determined the asymptotic behaviours of all solutions as the theory of planar competitive maps guarantees that every positive solution converges, see, e.g. Liu and Elaydi [23] and Smith [31]. For $n\ge 3$, there are only some partial results. For instance, Ruiz-Herrera [30], Chow and Hsieh [5] and Ackleh et al. [1] show that in the competitive Leslie–Gower model, the competitive exclusion principle holds if only one species has the largest carrying capacity. So every solution converges to a boundary equilibrium which is globally stable. Recently, Balreira et al. [3] gave a general result on higher dimensional monotone maps that guarantees the global asymptotic stability of an interior equilibrium of system (19(19) $x_i(t+1)=\frac{{\lambda }_i{x}_i(t)}{1+x_i(t)+\sum _{j\ne i}{c}_{ij}{x}_j(t)},1\le i\le n.$(19) ) with n = 3 and all $c_{ij}=c<({\lambda }_3-1)/({\lambda }_1+{\lambda }_2-{\lambda }_3-1)$. Chow and Palmer [6] showed that when n = 3 and $c_{ij}=c\gg 1$, the unique interior equilibrium of system (19(19) $x_i(t+1)=\frac{{\lambda }_i{x}_i(t)}{1+x_i(t)+\sum _{j\ne i}{c}_{ij}{x}_j(t)},1\le i\le n.$(19) ), if exists, is a saddle with one dimensional stable manifold. They conjectured that every positive solution converges. We are thus motivated to find some special systems like (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ) and (18(18) $x_i(t+1)=\frac{a{x}_i(t)}{1+x_i(t)+c\sum _{j\ne i}{x}_j(t)},1\le i\le n,$(18) ) such that this conjecture holds.

It is clear that assumption (A1) implies both ${\mathbb{R}}_{+}^n$ and $\mathrm{int}{\mathbb{R}}_{+}^n$ are forward invariant for system (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ). The following lemma shows that any solution of system (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ) converges to neither $\mathbf{0}$ nor infinity under assumption (A3).

Lemma 3.1

Assume that (A1) and (A3) hold. Then for any $\mathit{x}\in {\mathbb{R}}_{+}^n\mathrm{\setminus }\{\mathbf{0}\}$, the omega-limit set $\omega (\mathit{x})$ of system (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ) is a compact subset of ${\mathbb{R}}_{+}^n\mathrm{\setminus }\{\mathbf{0}\}$.

Proof.

Let $K_i=\{\mathit{x}\in {\mathbb{R}}_{+}^n:x_i\le L\}$, $1\le i\le n$ and $\mathit{x}(t)$ be a solution of (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ). Remember L and $\lambda <1$ are defined in (15(15) $\text{there exist }L>0\text{ and }\lambda \in (0,1)\text{ such that }\prod _{\ell =1}^m{h}_{\ell }(x)<\lambda \text{ for all }x\ge L.$(15) ). Suppose that $\mathit{x}(t)\in {\mathbb{R}}_{+}^n\mathrm{\setminus }{K}_i$. Then $x_i(t)+c_{\ell }\sum _{j\ne i}{x}_j(t)\ge x_i(t)>L$ holds for all $1\le \ell \le m$. By (15(15) $\text{there exist }L>0\text{ and }\lambda \in (0,1)\text{ such that }\prod _{\ell =1}^m{h}_{\ell }(x)<\lambda \text{ for all }x\ge L.$(15) ) and (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ), we have $\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }\sum _{j\ne i}{x}_j(t))<\lambda <1\text{ and then }{x}_i(t+1)\le \lambda x_i(t).$ Thus, the solution $\mathit{x}(t)$ eventually enters $\bigcap _{i=1}^n{K}_i$. Since the right-hand side of (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ) is a continuous function of $\mathit{x}(t)$ and $\bigcap _{i=1}^n{K}_i$ is compact, the solution $\mathit{x}(t)$ is bounded. Thus, $\omega (\mathit{x}(0))$ is compact. Similarly, the first inequality in (A3) implies $\mathit{x}(t)$ with $\mathit{x}(0)\in {\mathbb{R}}_{+}^n\mathrm{\setminus }\{\mathbf{0}\}$ eventually enters the compact set $K=\bigcap _{i=1}^n{K}_i\cap \{\mathit{x}\in {\mathbb{R}}_{+}^n:ϵ\le \sum _{i=1}^n{x}_i\}$ with sufficiently small $ϵ>0$. Since the right-hand side of (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ) is a continuous function of $\mathit{x}(t)$ and all $h_{\ell }>0$ by (A1), we can conclude that $\omega (\mathit{x}(0))\subset {\mathbb{R}}_{+}\mathrm{\setminus }\{\mathbf{0}\}$.

Based on assumptions (A1) and (A2), we first show a preliminary lemma.

Lemma 3.2

Assume (A1) and (A2) hold. Then for any $b_{\ell }>0$ and $1\le \ell \le m$, both $x\prod _{\ell =1}^m{h}_{\ell }(x+b_{\ell })$ and $x\prod _{\ell =1}^m{h}_{\ell }(b_{\ell }x)$ are increasing for $x\in [0,\mathrm{\infty })$.

Proof.

Let $0<{\alpha }_{\ell }<1$ be given in (A2). Then $(x+b_{\ell }{)}^{{\alpha }_{\ell }}-x^{{\alpha }_{\ell }}$ is positive, decreasing in x as $\frac{\mathrm{d}}{\mathrm{d}x}((x+b_{\ell }{)}^{{\alpha }_{\ell }}-x^{{\alpha }_{\ell }})={\alpha }_{\ell }((x+b_{\ell }{)}^{{\alpha }_{\ell }-1}-x^{{\alpha }_{\ell }-1})\le 0.$ By (A2) and (A1), $(x+b_{\ell }{)}^{{\alpha }_{\ell }}{h}_{\ell }(x+b_{\ell })$ increases and $h_{\ell }(x+b_{\ell })$ decreases in x. Hence, $0\le x^{{\alpha }_{\ell }}{h}_{\ell }(x+b_{\ell })=(x+b_{\ell }{)}^{{\alpha }_{\ell }}{h}_{\ell }(x+b_{\ell })-((x+b_{\ell }{)}^{{\alpha }_{\ell }}-x^{{\alpha }_{\ell }})h_{\ell }(x+b_{\ell })$ is increasing in x. Using $\sum _{\ell =1}^m{\alpha }_{\ell }=1$, the first claim follows from $x\prod _{\ell =1}^m{h}_{\ell }(x+b_{\ell })=\prod _{\ell =1}^m(x^{{\alpha }_{\ell }}{h}_{\ell }(x+b_{\ell })).$ Similarly, $(b_{\ell }x{)}^{{\alpha }_{\ell }}{h}_{\ell }(b_{\ell }x)$ increases in x by (A2). The remaining claim follows from multiplying these functions together and using $\sum _{\ell =1}^m{\alpha }_{\ell }=1$ again.

Define $\mathit{c}=(c_1,c_2,\dots ,c_m{)}^{\mathrm{\top }}$ and $\mathbf{1}=(1,1,\dots ,1{)}^{\mathrm{\top }}$. Using Lemmas 3.1 and 3.2, we now show the following result on global dynamics of system (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ).

Theorem 3.3

Assume that (A1)–(A3) hold and $\mathit{x}(t)$ is a solution of system (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ) with $\mathit{x}(0)\in \mathrm{int}{\mathbb{R}}_{+}^n$. Let $M=\underset{1\le i\le n}{max}{x}_i(0)$ and $J=\{i:x_i(0)=M\}$.

1. If all $c_{\ell }\in [0,1]$ and $\mathit{c}\ne \mathbf{1}$, then $\underset{t\to \mathrm{\infty }}{lim}\mathit{x}(t)=\eta \mathbf{1}$, where η is uniquely determined by $\prod _{\ell =1}^m{h}_{\ell }((1+c_{\ell }(n-1))\eta )=1$.

2. If all $c_{\ell }\ge 1$ and $\mathit{c}\ne \mathbf{1}$, then $\underset{t\to \mathrm{\infty }}{lim}\mathit{x}(t)=\eta \sum _{j\in J}{\mathit{e}}_j$, where η is uniquely determined by $\prod _{\ell =1}^m{h}_{\ell }((1+c_{\ell }(|J|-1))\eta )=1$ and ${\mathit{e}}_j$ is the jth unit vector. In particular, $\underset{t\to \mathrm{\infty }}{lim}{x}_i(t)=0$ for $i\notin J$.

3. If all $c_{\ell }=1$, then $\underset{t\to \mathrm{\infty }}{lim}\mathit{x}(t)=\eta \mathit{x}(0)$, where η is uniquely determined by $\prod _{\ell =1}^m{h}_{\ell }(\eta \sum _{j=1}^n{x}_j(0))=1$.

Proof.

Note that the existence and uniqueness of η in (a) –(c) is due to assumptions (A1) and (A3). Define $S_i(t)=\sum _{j\ne i}{x}_j(t)$. By symmetry, we may assume $x_1(0)=M$.

Part (a). We show by induction on $t\in {\mathbb{Z}}_{+}=\{0,1,2,\dots \}$ that for $2\le i\le n$, (20) $\frac{x_i(t)}{x_1(t)}\le 1\text{ and is increasing in }t.\text{ Thus }\underset{t\to \mathrm{\infty }}{lim}{\displaystyle \frac{x_i(t)}{x_1(t)}=r_i\in (0,1]\text{ exists.}}$(20) Assume that all $x_i(t)\le x_1(t)$ hold for some $t\in {\mathbb{Z}}_{+}$. Note that all $x_i(0)\le M=x_1(0)$ by assumption. Then certainly $S_i(t)\ge S_1(t)$. Using (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ) and the first claim in Lemma 3.2, we obtain $x_i(t+1)/x_1(t+1)\le 1$ as follows: $\begin{array}{rl}{x}_i(t+1)& =x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{S}_i(t))\le x_1(t)\prod _{\ell =1}^m{h}_{\ell }(x_1(t)+c_{\ell }{S}_i(t))\\ & \le x_1(t)\prod _{\ell =1}^m{h}_{\ell }(x_1(t)+c_{\ell }{S}_1(t))=x_1(t+1),\end{array}$ where in the last inequality (A1) and $S_i(t)\ge S_1(t)$ are used.

Since all $c_{\ell }\le 1$, $x_i(t)+c_{\ell }{S}_i(t)\le x_1(t)+c_{\ell }{S}_1(t)$ can be verified as follows: (21) $(x_i(t)+c_{\ell }{S}_i(t))-(x_1(t)+c_{\ell }{S}_1(t))=(1-c_{\ell })(x_i(t)-x_1(t))\le 0.$(21) Using (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ), (21(21) $(x_i(t)+c_{\ell }{S}_i(t))-(x_1(t)+c_{\ell }{S}_1(t))=(1-c_{\ell })(x_i(t)-x_1(t))\le 0.$(21) ) and (A1) again, we obtain (22) $\frac{x_i(t+1)}{x_i(t)}=\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{S}_i(t))\ge \prod _{\ell =1}^m{h}_{\ell }(x_1(t)+c_{\ell }{S}_1(t))=\frac{x_1(t+1)}{x_1(t)}.$(22) Thus $x_i(t)/x_1(t)\le x_i(t+1)/x_1(t+1)$ and (20(20) $\frac{x_i(t)}{x_1(t)}\le 1\text{ and is increasing in }t.\text{ Thus }\underset{t\to \mathrm{\infty }}{lim}{\displaystyle \frac{x_i(t)}{x_1(t)}=r_i\in (0,1]\text{ exists.}}$(20) ) is verified by induction.

Equation (20(20) $\frac{x_i(t)}{x_1(t)}\le 1\text{ and is increasing in }t.\text{ Thus }\underset{t\to \mathrm{\infty }}{lim}{\displaystyle \frac{x_i(t)}{x_1(t)}=r_i\in (0,1]\text{ exists.}}$(20) ) and Lemma 3.1 imply that $\omega (\mathit{x}(0))\subseteq \{\mathit{x}=s(r_1,r_2,r_3,\dots ,r_n{)}^{\mathrm{\top }}:s\in (0,\mathrm{\infty })\}=Y$, where $r_1=1$ so that (20(20) $\frac{x_i(t)}{x_1(t)}\le 1\text{ and is increasing in }t.\text{ Thus }\underset{t\to \mathrm{\infty }}{lim}{\displaystyle \frac{x_i(t)}{x_1(t)}=r_i\in (0,1]\text{ exists.}}$(20) ) is valid for i = 1 as well. On the half-line Y, system (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ) is reduced to n one-dimensional equations: (23) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)(1+\frac{c_{\ell }}{r_i}\sum _{j\ne i}{r}_j)).$(23) Note that, by Lemma 3.2 and (A1), every solution on Y converges to the point $({\eta }_1,{\eta }_2,\dots ,{\eta }_n{)}^{\mathrm{\top }}$, where ${\eta }_i>0$ is uniquely determined by $\prod _{\ell =1}^m{h}_{\ell }({\eta }_i(1+(c_{\ell }/r_i)\sum _{j\ne i}{r}_j))=1$. Since $\omega (\mathit{x}(0))\subseteq {\mathbb{R}}_{+}^n\mathrm{\setminus }\{\mathbf{0}\}$ is compact by Lemma 3.1, $\mathit{x}(t)$ is bounded for $t\ge 0$. The boundedness ensures that $\omega (\mathit{x}(0))$ is also nonempty and invariantly connected (e.g. see Theorem 5.2, LaSalle [21]). Since $\{({\eta }_1,{\eta }_2,\dots ,{\eta }_n{)}^{\mathrm{\top }}\}$ is a unique nonempty invariantly connected set in Y, we can conclude that $\omega (\mathit{x}(0))=\{({\eta }_1,{\eta }_2,\dots ,{\eta }_n{)}^{\mathrm{\top }}\}$, i.e. for $1\le i\le n$, (24) $\underset{t\to \mathrm{\infty }}{lim}{x}_i(t)={\eta }_i.$(24) We will see below that $r_i$ and thus ${\eta }_i$ are independent of the initial condition.

Write ${\eta }_1=\eta$. Then ${\eta }_i=r_i\eta$ by (20(20) $\frac{x_i(t)}{x_1(t)}\le 1\text{ and is increasing in }t.\text{ Thus }\underset{t\to \mathrm{\infty }}{lim}{\displaystyle \frac{x_i(t)}{x_1(t)}=r_i\in (0,1]\text{ exists.}}$(20) ). Remember $r_1=1$. The defining equations for ${\eta }_1$ and ${\eta }_i$ above can be rewritten as (25) $\prod _{\ell =1}^m{h}_{\ell }\left(\eta (r_1+c_{\ell }\sum _{j\ne 1}{r}_j)\right)=1\text{ and }\prod _{\ell =1}^m{h}_{\ell }\left(\eta (r_i+c_{\ell }\sum _{j\ne i}{r}_j)\right)=1.$(25) Since all $c_{\ell }\le 1$ by assumption and $r_i\le 1=r_1$ by (20(20) $\frac{x_i(t)}{x_1(t)}\le 1\text{ and is increasing in }t.\text{ Thus }\underset{t\to \mathrm{\infty }}{lim}{\displaystyle \frac{x_i(t)}{x_1(t)}=r_i\in (0,1]\text{ exists.}}$(20) ), we find that (26) $(r_1+c_{\ell }\sum _{j\ne 1}{r}_j)-(r_i+c_{\ell }{\displaystyle \sum _{j\ne i}{r}_j)=(1-c_{\ell })(r_1-r_i)\ge 0\text{ for all }\ell \le m,i\le n.}$(26) Because each $h_{\ell }$ is strictly decreasing by (A1), equality in (26(26) $(r_1+c_{\ell }\sum _{j\ne 1}{r}_j)-(r_i+c_{\ell }{\displaystyle \sum _{j\ne i}{r}_j)=(1-c_{\ell })(r_1-r_i)\ge 0\text{ for all }\ell \le m,i\le n.}$(26) ) holds by (25(25) $\prod _{\ell =1}^m{h}_{\ell }\left(\eta (r_1+c_{\ell }\sum _{j\ne 1}{r}_j)\right)=1\text{ and }\prod _{\ell =1}^m{h}_{\ell }\left(\eta (r_i+c_{\ell }\sum _{j\ne i}{r}_j)\right)=1.$(25) ). Then all $r_i=r_1=1$ and thus, all ${\eta }_i=\eta$ as some $c_{\ell }<1$ by assumption. We conclude from (24(24) $\underset{t\to \mathrm{\infty }}{lim}{x}_i(t)={\eta }_i.$(24) ) and (25(25) $\prod _{\ell =1}^m{h}_{\ell }\left(\eta (r_1+c_{\ell }\sum _{j\ne 1}{r}_j)\right)=1\text{ and }\prod _{\ell =1}^m{h}_{\ell }\left(\eta (r_i+c_{\ell }\sum _{j\ne i}{r}_j)\right)=1.$(25) ) that $\underset{t\to \mathrm{\infty }}{lim}{x}_i(t)=\eta$ for $1\le i\le n$ and η is uniquely determined by $\prod _{\ell =1}^m{h}_{\ell }(\eta (1+c_{\ell }(n-1)))=1$. This verifies the assertion in (a).

Part (b). Since $x_1(0)=M$ by assumption, we have $x_i(0)=x_1(0)$ for $i\in J$ and $x_i(0)<x_1(0)$ for $i\notin J$. As was done to show (20(20) $\frac{x_i(t)}{x_1(t)}\le 1\text{ and is increasing in }t.\text{ Thus }\underset{t\to \mathrm{\infty }}{lim}{\displaystyle \frac{x_i(t)}{x_1(t)}=r_i\in (0,1]\text{ exists.}}$(20) ), we show by induction on t that for $2\le i\le n$, (27) $\frac{x_i(t)}{x_1(t)}\le 1\text{ and is decreasing in }t\text{ with }\frac{x_i(t)}{x_1(t)}=1\text{ for }i\in J.$(27) As a consequence, there exist constants $r_i\in [0,1]$ with $r_1=1$ such that (28) $\underset{t\to \mathrm{\infty }}{lim}\frac{x_i(t)}{x_1(t)}=r_i\text{ and }{r}_i=1\text{ if and only if }i\in J.$(28) Assume that all $x_i(t)\le x_1(t)$ hold for some $t\in {\mathbb{Z}}_{+}$ with equality held for $i\in J$. The inequality $x_i(t+1)\le x_1(t+1)$ can be verified exactly as in the formula before (21(21) $(x_i(t)+c_{\ell }{S}_i(t))-(x_1(t)+c_{\ell }{S}_1(t))=(1-c_{\ell })(x_i(t)-x_1(t))\le 0.$(21) ). Since $c_{\ell }\ge 1$, the inequality in (21(21) $(x_i(t)+c_{\ell }{S}_i(t))-(x_1(t)+c_{\ell }{S}_1(t))=(1-c_{\ell })(x_i(t)-x_1(t))\le 0.$(21) ) is reversed with equality held for $i\in J$. So is the inequality in (22(22) $\frac{x_i(t+1)}{x_i(t)}=\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{S}_i(t))\ge \prod _{\ell =1}^m{h}_{\ell }(x_1(t)+c_{\ell }{S}_1(t))=\frac{x_1(t+1)}{x_1(t)}.$(22) ). Hence, $x_i(t+1)/x_1(t+1)\le x_i(t)/x_1(t)$ with equality held for $i\in J$. This proves (27(27) $\frac{x_i(t)}{x_1(t)}\le 1\text{ and is decreasing in }t\text{ with }\frac{x_i(t)}{x_1(t)}=1\text{ for }i\in J.$(27) ) by induction and thus (28(28) $\underset{t\to \mathrm{\infty }}{lim}\frac{x_i(t)}{x_1(t)}=r_i\text{ and }{r}_i=1\text{ if and only if }i\in J.$(28) ) is verified.

Following the same arguments in the proof of Part (a), we have $\omega (\mathit{x}(0))\subseteq \{\mathit{x}=s(r_1,r_2,r_3,\dots ,r_n{)}^{\mathrm{\top }}:s\in (0,\mathrm{\infty })\}$. Let $K=\{i\notin J:r_i>0\}$. On the half-line above, system (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ) is reduced to $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)(1+\frac{c_{\ell }}{r_i}\sum _{j\ne i}{r}_j))\text{ for }i\in J\cup K,$ and $lim{x}_i(t)=0$ for $i\notin J\cup K$ as $\mathit{x}(t)$ is bounded by Lemma 3.1 and $r_i=0$ for $i\notin J\cup K$. Similar to (24(24) $\underset{t\to \mathrm{\infty }}{lim}{x}_i(t)={\eta }_i.$(24) ), we can get from Lemma 3.2 and (28(28) $\underset{t\to \mathrm{\infty }}{lim}\frac{x_i(t)}{x_1(t)}=r_i\text{ and }{r}_i=1\text{ if and only if }i\in J.$(28) ) that (29) $\underset{t\to \mathrm{\infty }}{lim}{x}_i(t)={\eta }_i\text{ for }1\le i\le n.$(29) Here, ${\eta }_i=\eta$ for $i\in J$, ${\eta }_i=r_i\eta <\eta$ for $i\in K$ and ${\eta }_i=0$ for $i\notin J\cup K$. Moreover, the defining equations for ${\eta }_1=\eta$ and ${\eta }_i=\eta r_i$, $i\in K$ are given by (30) $\prod _{\ell =1}^m{h}_{\ell }\left(\eta (r_1+c_{\ell }\sum _{j\ne 1}{r}_j)\right)=1\text{ and }\prod _{\ell =1}^m{h}_{\ell }\left(\eta (r_i+c_{\ell }\sum _{j\ne i}{r}_j)\right)=1.$(30) We claim $K=\mathrm{\varnothing }$. Then the conclusion in Part (b) follows from (29(29) $\underset{t\to \mathrm{\infty }}{lim}{x}_i(t)={\eta }_i\text{ for }1\le i\le n.$(29) ) with η uniquely determined by $\prod _{\ell =1}^m{h}_{\ell }(\eta (1+c_{\ell }(|J|-1)))=1$ as shown in the first equality in (31(31) $1=\prod _{\ell =1}^m{h}_{\ell }\left(\eta (1+c_{\ell }(|J|-1+\sum _{j\in K}{r}_j))\right)=\prod _{\ell =1}^m{h}_{\ell }\left(\eta (r_i+c_{\ell }(|J|+\sum _{\begin{array}{c}j\ne i\\ j\in K\end{array}}{r}_j))\right).$(31) ).

Suppose the contrary that $K\ne \mathrm{\varnothing }$. Using $r_j=r_1=1$ for $j\in J$ and $r_j=0$ for $j\notin J\cup K$, we may rewrite (30(30) $\prod _{\ell =1}^m{h}_{\ell }\left(\eta (r_1+c_{\ell }\sum _{j\ne 1}{r}_j)\right)=1\text{ and }\prod _{\ell =1}^m{h}_{\ell }\left(\eta (r_i+c_{\ell }\sum _{j\ne i}{r}_j)\right)=1.$(30) ) as (31) $1=\prod _{\ell =1}^m{h}_{\ell }\left(\eta (1+c_{\ell }(|J|-1+\sum _{j\in K}{r}_j))\right)=\prod _{\ell =1}^m{h}_{\ell }\left(\eta (r_i+c_{\ell }(|J|+\sum _{\begin{array}{c}j\ne i\\ j\in K\end{array}}{r}_j))\right).$(31) Since all $c_{\ell }\ge 1$ by assumption and $r_i<1=r_1$ for $i\in K$ by (28(28) $\underset{t\to \mathrm{\infty }}{lim}\frac{x_i(t)}{x_1(t)}=r_i\text{ and }{r}_i=1\text{ if and only if }i\in J.$(28) ), (32) $(1+c_{\ell }(|J|-1+\sum _{j\in K}{r}_j))-(r_i+c_{\ell }(|J|+\sum _{\begin{array}{c}j\ne i\\ j\in K\end{array}}{r}_j))=(1-c_{\ell })(1-r_i)\le 0$(32) for all $1\le \ell \le m$. Because all $h_{\ell }$ decreases strictly by (A1), the equality in (32(32) $(1+c_{\ell }(|J|-1+\sum _{j\in K}{r}_j))-(r_i+c_{\ell }(|J|+\sum _{\begin{array}{c}j\ne i\\ j\in K\end{array}}{r}_j))=(1-c_{\ell })(1-r_i)\le 0$(32) ) holds by (31(31) $1=\prod _{\ell =1}^m{h}_{\ell }\left(\eta (1+c_{\ell }(|J|-1+\sum _{j\in K}{r}_j))\right)=\prod _{\ell =1}^m{h}_{\ell }\left(\eta (r_i+c_{\ell }(|J|+\sum _{\begin{array}{c}j\ne i\\ j\in K\end{array}}{r}_j))\right).$(31) ). This leads to a contradiction as $\mathit{c}\ne \mathbf{1}$ by assumption and $r_i<1$ for $i\in K$.

Part (c). When all $c_{\ell }=1$, all $x_i(t)+c_{\ell }{S}_i(t)$ equal $\sum _{j=1}^n{x}_j(t)$. The inequalities in both (21(21) $(x_i(t)+c_{\ell }{S}_i(t))-(x_1(t)+c_{\ell }{S}_1(t))=(1-c_{\ell })(x_i(t)-x_1(t))\le 0.$(21) ) and (22(22) $\frac{x_i(t+1)}{x_i(t)}=\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{S}_i(t))\ge \prod _{\ell =1}^m{h}_{\ell }(x_1(t)+c_{\ell }{S}_1(t))=\frac{x_1(t+1)}{x_1(t)}.$(22) ) become equalities. So we have $\frac{x_i(t+1)}{x_1(t+1)}=\frac{x_i(t)}{x_1(t)}=\frac{x_i(t-1)}{x_1(t-1)}=\cdots =\frac{x_i(0)}{x_1(0)}.$ That is, (33) $x_j(t)=x_1(t)\frac{x_j(0)}{x_1(0)}\text{ for all }2\le j\le n\text{ and }t\ge 0.$(33) Therefore, system (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ) for i = 1 can be written as (34) $x_1(t+1)=x_1(t)\prod _{\ell =1}^m{h}_{\ell }(x_1(t)\frac{\sum _{1\le j\le n}{x}_j(0)}{x_1(0)}).$(34) Then $\underset{t\to \mathrm{\infty }}{lim}{x}_1(t)=\eta x_1(0)$ with η uniquely determined by $\prod _{\ell =1}^m{h}_{\ell }(\eta \sum _{j=1}^n{x}_j(0))=1$. By (33(33) $x_j(t)=x_1(t)\frac{x_j(0)}{x_1(0)}\text{ for all }2\le j\le n\text{ and }t\ge 0.$(33) ), $\underset{t\to \mathrm{\infty }}{lim}{x}_i(t)=\eta x_i(0)$ for $2\le i\le n$ as claimed. The proof is complete.

As a consequence, we have the following result for system (5(5) $x_i(t+1)=x_i(t)h(x_i(t)+c\sum _{j\ne i}{x}_j(t)),1\le i\le n,$(5) ).

Corollary 3.4

Assume h satisfies (A1)${}^{\mathrm{\prime }}$-(A3)${}^{\mathrm{\prime }}$ and $\mathit{x}(t)$ is a solution of (5(5) $x_i(t+1)=x_i(t)h(x_i(t)+c\sum _{j\ne i}{x}_j(t)),1\le i\le n,$(5) ) with $\mathit{x}(0)\in \mathrm{int}{\mathbb{R}}_{+}^n$. Let $M=\underset{1\le i\le n}{max}{x}_i(0)$ and $J=\{i:x_i(0)=M\}$.

1. For $0\le c<1$, $\underset{t\to \mathrm{\infty }}{lim}\mathit{x}(t)=\eta \mathbf{1}$ with η uniquely determined by $h((1+c(n-1))\eta )=1$.

2. For c>1, $\underset{t\to \mathrm{\infty }}{lim}\mathit{x}(t)=\eta \sum _{j\in J}{\mathit{e}}_j$ with η uniquely determined by $h((1+c(|J|-1))\eta )=1$. In particular, $\underset{t\to \mathrm{\infty }}{lim}{x}_i(t)=0$ for $i\notin J$.

3. For c = 1, $\underset{t\to \mathrm{\infty }}{lim}\mathit{x}(t)=\eta \mathit{x}(0)$ with η uniquely determined by $h(\eta \sum _{j=1}^n{x}_j(0))=1$.

We remark that since $x_i(0)=0$ implies $x_i(t)=0$ for all $t\in {\mathbb{Z}}_{+}$, the results above can be easily extended to $\mathit{x}(0)\in {\mathbb{R}}_{+}^n$. Depending on c<1, c = 1 or c>1, the asymptotic behaviour of system (5(5) $x_i(t+1)=x_i(t)h(x_i(t)+c\sum _{j\ne i}{x}_j(t)),1\le i\le n,$(5) ) is quite different. Yet, both Theorem 3.3 and Corollary 3.4 show that every positive solution converges to some equilibrium.

4. Asymptotic stability of Kolmogorov difference equations

We discuss in this section the local asymptotic stability of some equilibria in system (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ). For this purpose, we have to assume that besides (A1)–(A3), all $h_{\ell }$ in system (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ) are differentiable. Note that $h_{\ell }^{\mathrm{\prime }}\le 0$ by assumption (A1). Theorem 3.3 (a) says that $E^{\ast }={\eta }^{\ast }\mathbf{1}$ is the unique interior equilibrium with ${\eta }^{\ast }$ uniquely determined by (35) $\prod _{\ell =1}^m{h}_{\ell }(N_{\ell })=1,\text{ where }{N}_{\ell }=(1+c_{\ell }(n-1)){\eta }^{\ast }.$(35) Furthermore, Theorem 3.3(b) says that if $x_i(0)>\underset{i\ne j}{max}{x}_j(0)$, then $|J|=1$ and the solution $\mathit{x}(t)$ of (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ) converges to the single-species equilibrium $E_i=\eta {\mathit{e}}_i$, where η is uniquely determined by (36) $\prod _{\ell =1}^m{h}_{\ell }\left(\eta \right)=1.$(36) Concerning the local asymptotic stability of these equilibria, we have the following result.

Theorem 4.1

Besides (A1) –(A3), we assume all $h_{\ell }^{\mathrm{\prime }}<0$ on $(0,\mathrm{\infty })$.

1. If all $c_{\ell }\in [0,1]$ and $\mathit{c}\ne \mathbf{1}$, then $E^{\ast }={\eta }^{\ast }\mathbf{1}$ is locally asymptotically stable.

2. If all $c_{\ell }\ge 1$ and $\mathit{c}\ne \mathbf{1}$, then each $E_i=\eta {\mathit{e}}_i$, $1\le i\le n$, is locally asymptotically stable.

Proof.

Part (a). Let $B(\mathit{x})$ be the Jacobian matrix of system (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ) evaluated at $\mathit{x}$. Using (35(35) $\prod _{\ell =1}^m{h}_{\ell }(N_{\ell })=1,\text{ where }{N}_{\ell }=(1+c_{\ell }(n-1)){\eta }^{\ast }.$(35) ), $B(E^{\ast })$ is an $n\times n$ matrix whose diagonal and off-diagonal entries are, respectively, given by $1+{\eta }^{\ast }\sum _{k=1}^m\prod _{\ell \ne k}{h}_{\ell }(N_{\ell })h_k^{\prime }(N_k)\text{ and }{\eta }^{\ast }\sum _{k=1}^m\prod _{\ell \ne k}{h}_{\ell }(N_{\ell })h_k^{\prime }(N_k)c_k,$ where $N_{\ell }$ are defined in (35(35) $\prod _{\ell =1}^m{h}_{\ell }(N_{\ell })=1,\text{ where }{N}_{\ell }=(1+c_{\ell }(n-1)){\eta }^{\ast }.$(35) ). Being a circulant matrix [12], its eigenvalues are ${\lambda }_p=1+{\eta }^{\ast }\sum _{k=1}^m\prod _{\ell \ne k}{h}_{\ell }(N_{\ell })h_k^{\prime }(N_k)+{\eta }^{\ast }\sum _{k=1}^m\prod _{\ell \ne k}{h}_{\ell }(N_{\ell })h_k^{\prime }(N_k)c_k\sum _{j=1}^{n-1}{e}^{(2\pi i/n)jp}$ for $0\le p\le n-1$. Using $(1+c_k(n-1)){\eta }^{\ast }=N_k$ and all $h_{\ell }>0,h_{\ell }^{\mathrm{\prime }}<0$, we get (37) ${\lambda }_0=1+\sum _{k=1}^m\prod _{\ell \ne k}{h}_{\ell }(N_{\ell })h_k^{\prime }(N_k)N_k<1.$(37) Because all $c_{\ell }\in [0,1]$ with some $c_{\ell }<1$ by assumption and $\sum _{j=1}^{n-1}{e}^{(2\pi i/n)jp}=-1$ for $p\ge 1$, (38) ${\lambda }_p=1+\sum _{k=1}^m\prod _{\ell \ne k}{h}_{\ell }(N_{\ell })h_k^{\prime }(N_k)(1-c_k){\eta }^{\ast }<1\text{ for }1\le p\le n-1.$(38) We claim that for $1\le p\le n-1$, (39) ${\lambda }_p\ge {\lambda }_0\ge 0.$(39) Together with (37(37) ${\lambda }_0=1+\sum _{k=1}^m\prod _{\ell \ne k}{h}_{\ell }(N_{\ell })h_k^{\prime }(N_k)N_k<1.$(37) ) and (38(38) ${\lambda }_p=1+\sum _{k=1}^m\prod _{\ell \ne k}{h}_{\ell }(N_{\ell })h_k^{\prime }(N_k)(1-c_k){\eta }^{\ast }<1\text{ for }1\le p\le n-1.$(38) ), we then have all ${\lambda }_p\in [0,1)$ and thus $E^{\ast }$ is asymptotically stable. Using $N_k=(1+c_k(n-1)){\eta }^{\ast }$, the first inequality above follows easily as: ${\lambda }_p-{\lambda }_0=-\sum _{k=1}^m\prod _{\ell \ne k}{h}_{\ell }(N_{\ell })h_k^{\prime }(N_k)c_{k}n{\eta }^{\ast }\ge 0\text{ for }p\ge 1.$ It remains to show ${\lambda }_0\ge 0$. By assumption (A2), $0\le \frac{\mathrm{d}}{\mathrm{d}x}(x^{{\alpha }_k}{h}_k(x)){|}_{x=N_k}={\alpha }_k{N}_k^{{\alpha }_k-1}{h}_k(N_k)+N_k^{{\alpha }_k}{h}_k^{\prime }(N_k),$ which yields $h_k^{\prime }(N_k)N_k\ge -{\alpha }_k{h}_k(N_k)$. Applying this inequality to (37(37) ${\lambda }_0=1+\sum _{k=1}^m\prod _{\ell \ne k}{h}_{\ell }(N_{\ell })h_k^{\prime }(N_k)N_k<1.$(37) ), we get ${\lambda }_0\ge 1+\sum _{k=1}^m\prod _{\ell \ne k}{h}_{\ell }(N_{\ell })(-{\alpha }_k)h_k(N_k)=1+\sum _{k=1}^m(-{\alpha }_k)\prod _{\ell =1}^m{h}_{\ell }(N_{\ell })=0$ as $\prod _{\ell =1}^m{h}_{\ell }(N_{\ell })=1$ by (35(35) $\prod _{\ell =1}^m{h}_{\ell }(N_{\ell })=1,\text{ where }{N}_{\ell }=(1+c_{\ell }(n-1)){\eta }^{\ast }.$(35) ) and $\sum _{k=1}^m{\alpha }_k=1$ by (A3).

Part (b). By symmetry, it suffices to consider $E_1=\eta {\mathit{e}}_1$. It is straightforward to show that $B(E_1)=(b_{ij})$ satisfies $b_{ij}=0$ for $j\ne i\ge 2$. Moreover, (36(36) $\prod _{\ell =1}^m{h}_{\ell }\left(\eta \right)=1.$(36) ) implies that (40) $b_{11}=1+\eta \sum _{k=1}^m\prod _{\ell \ne k}{h}_{\ell }(\eta )h_k^{\prime }(\eta )\text{ and }{b}_{ii}=\prod _{\ell =1}^m{h}_{\ell }(c_{\ell }\eta )\text{ for }2\le i\le n.$(40) Being an upper triangular matrix, $\{b_{ii}:1\le i\le n\}$ are eigenvalues of $B(E_1)$. Because all $c_{\ell }\ge 1$ with some $c_{\ell }>1$ and $0<h_{\ell }$ decreases strictly by (A1), we get from (36(36) $\prod _{\ell =1}^m{h}_{\ell }\left(\eta \right)=1.$(36) ) that (41) $0\le \prod _{\ell =1}^m{h}_{\ell }(c_{\ell }\eta )<\prod _{\ell =1}^m{h}_{\ell }(\eta )=1.$(41) By Lemma 3.2, $x\prod _{\ell =1}^m{h}_{\ell }(x)$ is increasing. Therefore, (42) $0\le \frac{\mathrm{d}}{\mathrm{d}x}(x\prod _{\ell =1}^m{h}_{\ell }(x)){|}_{x=\eta }=1+\eta \sum _{k=1}^m\prod _{\ell \ne k}{h}_{\ell }(\eta )h_k^{\prime }(\eta )<1,$(42) where we have used (36(36) $\prod _{\ell =1}^m{h}_{\ell }\left(\eta \right)=1.$(36) ) and the assumption that all $h_{\ell }>0$, $h_{\ell }^{\mathrm{\prime }}<0$. Combining together (40(40) $b_{11}=1+\eta \sum _{k=1}^m\prod _{\ell \ne k}{h}_{\ell }(\eta )h_k^{\prime }(\eta )\text{ and }{b}_{ii}=\prod _{\ell =1}^m{h}_{\ell }(c_{\ell }\eta )\text{ for }2\le i\le n.$(40) )–(42(42) $0\le \frac{\mathrm{d}}{\mathrm{d}x}(x\prod _{\ell =1}^m{h}_{\ell }(x)){|}_{x=\eta }=1+\eta \sum _{k=1}^m\prod _{\ell \ne k}{h}_{\ell }(\eta )h_k^{\prime }(\eta )<1,$(42) ), we obtain that all eigenvalues $b_{ii}$ lie in $[0,1)$. The assertion that $E_1=\eta {\mathit{e}}_1$ is asymptotically stable is verified.

Note that functions $h_{\ell }$ defined in (16(16) $h_{\ell }(x)={({\beta }_{\ell }+\frac{a_{\ell }}{1+x})}^{{\alpha }_{\ell }},1\le \ell \le m,$(16) ) satisfy all the assumptions in Theorem 4.1. By combining Theorem 4.1 with Theorem 3.3, we can conclude that, under the assumptions (A1) –(A3) and all $h_{\ell }^{\mathrm{\prime }}<0$, the interior equilibrium $E^{\ast }$ of system (14(14) $x_i(t+1)=x_i(t)\prod _{\ell =1}^m{h}_{\ell }(x_i(t)+c_{\ell }{\displaystyle \sum _{j\ne i}{x}_j(t)}),$(14) ) is globally asymptotically stable in $\mathrm{int}{\mathbb{R}}_{+}^n$ if all $c_{\ell }\in [0,1]$ and $\mathit{c}\ne \mathbf{1}$. Furthermore, every single-species equilibrium $E_i=\eta {\mathit{e}}_i$, $1\le i\le n$, is globally asymptotically stable in $\{\mathit{x}\in \mathrm{int}{\mathbb{R}}_{+}^n:x_i>\underset{j\ne i}{max}{x}_j\}$ if all $c_{\ell }\ge 1$ and $\mathit{c}\ne \mathbf{1}$.

5. Global dynamics of semelparous Leslie matrix models

While assumptions (H1) and (H2) are sufficient for the dynamical consistency relation shown in Theorem 2.2, they are too weak to get any global results for system (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ). We need to impose some extra conditions on the growth rate function g in system (2(2) $\begin{array}{rl}{x}_1(t+1)& =g(x_1(t),x_2(t),\dots ,x_n(t))x_1(t),\\ x_2(t+1)& =g(x_2(t),x_3(t),\dots ,x_1(t))x_2(t),\\ & ⋮\\ x_n(t+1)& =g(x_n(t),x_1(t),\dots ,x_{n-1}(t))x_n(t).\end{array}$(2) ):

(H3)	$g(\mathit{x})=h(x_1+c\sum _{j\ne 1}{x}_j)$ with function h satisfying (A1)${}^{\prime }$–(A3)${}^{\prime }$.

Then, Corollary 3.4 can be applied to system (2(2) $\begin{array}{rl}{x}_1(t+1)& =g(x_1(t),x_2(t),\dots ,x_n(t))x_1(t),\\ x_2(t+1)& =g(x_2(t),x_3(t),\dots ,x_1(t))x_2(t),\\ & ⋮\\ x_n(t+1)& =g(x_n(t),x_1(t),\dots ,x_{n-1}(t))x_n(t).\end{array}$(2) ) and global dynamics of system (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) follows immediately from Theorem 2.2.

Before stating the results, we note that ${\sigma }_i(\mathbf{0})=1$ by (H1) and $h(0)>1$ by (A3)${}^{\prime }$. Then (H2) and (H3) imply that (43) ${\sigma }_i(\mathit{u})={\mathcal{R}}_0^{-(1/n)}h(\frac{u_i}{d_i}+c\sum _{j\ne i}\frac{u_j}{d_j}),$(43) where function h satisfies (A1)${}^{\prime }$–(A3)${}^{\prime }$ and $h(0)={\mathcal{R}}_0^{1/n}=(s_1{s}_2\cdots s_n{)}^{1/n}>1$. For $j\ne i$, the constant c above measures the effect that the normalized density $u_j/d_j$ of age-class j has on the survival of age-class i. Thus competition intensities between age-classes are independent of age if the density effect is measured by the normalized population vector $(u_1/d_1,u_2/d_2,\dots ,u_n/d_n{)}^{\mathrm{\top }}$. For example, if ${\sigma }_i$ and g are given by ${\sigma }_i(\mathit{u})=\frac{1}{1+\frac{u_i}{d_i}+c\sum _{j\ne i}\frac{u_j}{d_j}}\mathrm{and}g(x)=\frac{{\mathcal{R}}_0^{1/n}}{1+x_1+c\sum _{j\ne 1}{x}_j},$ then (H2) and (H3) are satisfied with $h(x)={\mathcal{R}}_0^{1/n}/(1+x)$. With ${\mathcal{R}}_0=s_1{s}_2\cdots s_n$, we may use (43(43) ${\sigma }_i(\mathit{u})={\mathcal{R}}_0^{-(1/n)}h(\frac{u_i}{d_i}+c\sum _{j\ne i}\frac{u_j}{d_j}),$(43) ) to rewrite system (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) as (44) $\begin{array}{rl}{u}_1(t+1)& =s_n{\mathcal{R}}_0^{-(1/n)}h(\frac{u_n(t)}{d_n}+c\sum _{j\ne n}\frac{u_j(t)}{d_j})u_n(t),\\ u_2(t+1)& =s_1{\mathcal{R}}_0^{-(1/n)}h(\frac{u_1(t)}{d_1}+c\sum _{j\ne 1}\frac{u_j(t)}{d_j})u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\mathcal{R}}_0^{-(1/n)}h(\frac{u_{n-1}(t)}{d_{n-1}}+c\sum _{j\ne n-1}\frac{u_j(t)}{d_j})u_{n-1}(t).\end{array}$(44) Remember that matrices P and D are defined in (3(3) $P=\left(\begin{array}{ccccc}0& 0& \cdots & 0& 1\\ 1& 0& \cdots & 0& 0\\ 0& 1& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & 1& 0\end{array}\right)\mathrm{and}\mathit{x}(t)=\left(\begin{array}{c}{x}_1(t)\\ x_2(t)\\ ⋮\\ x_n(t)\end{array}\right).$(3) ) and (8(8) $y_i(t)=u_i(t)/d_i\text{ for }1\le i\le n,\text{ i.e., }D\mathit{y}(t)=\mathit{u}(t),$(8) ), respectively.

Theorem 5.1

Assume $s_i>0$ for $1\le i\le n$ and h satisfies (A1)${}^{\prime }$ –(A3)${}^{\prime }$ with $h(0)={\mathcal{R}}_0^{1/n}>1$. Let $\mathit{u}(t)$ be a solution of (44(44) $\begin{array}{rl}{u}_1(t+1)& =s_n{\mathcal{R}}_0^{-(1/n)}h(\frac{u_n(t)}{d_n}+c\sum _{j\ne n}\frac{u_j(t)}{d_j})u_n(t),\\ u_2(t+1)& =s_1{\mathcal{R}}_0^{-(1/n)}h(\frac{u_1(t)}{d_1}+c\sum _{j\ne 1}\frac{u_j(t)}{d_j})u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\mathcal{R}}_0^{-(1/n)}h(\frac{u_{n-1}(t)}{d_{n-1}}+c\sum _{j\ne n-1}\frac{u_j(t)}{d_j})u_{n-1}(t).\end{array}$(44) ) with $\mathit{u}(0)\in \mathrm{int}{\mathbb{R}}_{+}^n$. Define $M=\underset{1\le i\le n}{max}{u}_i(0)/d_i$ and $J=\{i:u_i(0)/d_i=M\}$.

1. If $0\le c<1$, then $\underset{t\to \mathrm{\infty }}{lim}\mathit{u}(t)=\hat{E}$, where $\hat{E}=\eta D\mathbf{1}$ with η uniquely determined by $h((1+c(n-1))\eta )=1$.

2. If c>1, then $\mathit{u}(t)$ converges to a cycle as $t\to \mathrm{\infty }$, i.e. $\omega (\mathit{u}(0))=\{{\mathit{p}}_1,{\mathit{p}}_2,\dots ,{\mathit{p}}_n\}$, where ${\mathit{p}}_i=\eta \sum _{j\in J}D{P}^{i-1}{\mathit{e}}_j$ for $1\le i\le n$ and η is uniquely determined by $h((1+c(|J|-1))\eta )=1$. In particular, $\mathit{u}(t)$ converges to the n-cycle $P^{\ast }\stackrel{\mathrm{def}}{=}\{\left(\begin{array}{c}\eta d_1\\ 0\\ ⋮\\ 0\end{array}\right),\left(\begin{array}{c}0\\ \eta d_2\\ ⋮\\ 0\end{array}\right),\dots ,\left(\begin{array}{c}0\\ 0\\ ⋮\\ \eta d_n\end{array}\right)\}$as $t\to \mathrm{\infty }$ if $|J|=1$, i.e. $\mathit{u}(0)\in \bigcup _{i=1}^n{V}_i$, where $V_i=\{\mathit{u}\in \mathrm{int}{\mathbb{R}}_{+}^n:u_i/d_i>\underset{j\ne i}{max}{u}_j/d_j\}$.

3. If c = 1, then $\mathit{u}(t)$ converges to a cycle as $t\to \mathrm{\infty }$, i.e. $\omega (\mathit{u}(0))=\{{\mathit{p}}_1,{\mathit{p}}_2,\dots ,{\mathit{p}}_n\}$, where ${\mathit{p}}_i=\eta D{P}^{i-1}{D}^{-1}\mathit{u}(0)$ for $1\le i\le n$ and η is uniquely determined by $h((\sum _{j=1}^n{u}_j(0)/d_j)\eta )=1$.

If h is also differentiable, then $\hat{E}$ is globally asymptotically stable in $\mathrm{int}{\mathbb{R}}_{+}^n$ if $0\le c<1$ and $P^{\ast }$ is globally asymptotically stable in $\bigcup _{i=1}^n{V}_i$ if c>1.

Proof.

Let $\mathit{x}(0)=D^{-1}\mathit{u}(0)$ and $\mathit{x}(t)$ be the solution of system (5(5) $x_i(t+1)=x_i(t)h(x_i(t)+c\sum _{j\ne i}{x}_j(t)),1\le i\le n,$(5) ) with the initial vector $\mathit{x}(0)$. Note that $\mathit{x}(0)$ is positive.

Part (a). By Corollary 3.4, $\mathit{x}(t)$ converges to $\eta \mathbf{1}$ as $t\to \mathrm{\infty }$. Then, Theorem 2.2 implies that $\mathit{u}(nk)\to \eta D\mathbf{1}$ as $k\to \mathrm{\infty }$. Since it holds for any positive $\mathit{u}(0)$, we conclude that $\mathit{u}(t)\to \eta D\mathbf{1}$ as $t\to \mathrm{\infty }$.

Part (b). Note that $M=max\{x_1(0),x_2(0),\dots ,x_n(0)\}$ and $J=\{i:x_i(0)=M\}$. As above, we get from Corollary 3.4 that $\mathit{x}(t)\to \eta \sum _{j\in J}{\mathit{e}}_j$ as $t\to \mathrm{\infty }$. By Theorem 2.2, $\mathit{u}(nk+m)\to \eta \sum _{j\in J}D{P}^m{\mathit{e}}_j$ as $k\to \mathrm{\infty }$ for every $0\le m\le n-1$. Thus $\mathit{u}(t)$ converges to the cycle $\{\eta \sum _{j\in J}D{\mathit{e}}_j,\eta \sum _{j\in J}DP{\mathit{e}}_j,,\dots ,\eta \sum _{j\in J}D{P}^{n-1}{\mathit{e}}_j\}$.

Part (c). By Corollary 3.4, $\mathit{x}(t)$ converges to $\eta \mathit{x}(0)$ as $t\to \mathrm{\infty }$. Using Theorem 2.2, $\mathit{u}(nk+m)\to \eta D{P}^m\mathit{x}(0)=\eta D{P}^m{D}^{-1}\mathit{u}(0)$ as $k\to \mathrm{\infty }$ for every $0\le m\le n-1$. Thus $\mathit{u}(t)$ converges to the cycle $\{\eta \mathit{u}(0),\eta DP{D}^{-1}\mathit{u}(0),\dots ,\eta D{P}^{n-1}{D}^{-1}\mathit{u}(0)\}$.

The assertion on the asymptotic stability follows from Theorem 4.1.

Note that the set $\mathrm{int}{\mathbb{R}}_{+}^n\mathrm{\setminus }\bigcup _{i=1}^n{V}_i$ has measure zero as it is a subset of $\bigcup _{1\le i\ne j\le n}{W}_{ij}$, where $W_{ij}=\{\mathit{u}\in {\mathbb{R}}_{+}^n:\frac{u_i}{d_i}=\frac{u_j}{d_j}\}\text{ for }1\le i\ne j\le n.$ Therefore, we find that generic solutions of (1(1) $\begin{array}{rl}{u}_1(t+1)& =s_n{\sigma }_n(u_1(t),u_2(t),\dots ,u_n(t))u_n(t),\\ u_2(t+1)& =s_1{\sigma }_1(u_1(t),u_2(t),\dots ,u_n(t))u_1(t),\\ & ⋮\\ u_n(t+1)& =s_{n-1}{\sigma }_{n-1}(u_1(t),u_2(t),\dots ,u_n(t))u_{n-1}(t),\end{array}$(1) ) converge to the n-cycle $P^{\ast }$ if c>1.

6. Concluding remarks

This paper provides a class of semelparous Leslie matrix models that are dynamically consistent with a certain system of Kolmogorov difference equations with cyclic symmetry. For some special class of the latter, we can determine its global dynamics. Then using the dynamical consistency established above, we obtain in Theorem 5.1 a class of semelparous Leslie matrix models that has, generically, either a globally asymptotically stable positive equilibrium or a globally asymptotically stable n-cycle. In Theorem 5.1, a strong assumption is imposed on the survival probabilities of the models. For instance, it is required that the competition intensities between age-classes are independent of age when the density effect is measured by suitably normalized population densities. It is shown that if the competition intensity between-age-class is larger than that of within-age-class, i.e. the case c>1 in Theorem 5.1, then the n-cycle associated with the periodic behaviour in periodical insects are globally asymptotically stable. It is also shown that if the situation is reversed, i.e. the case c<1 in Theorem 5.1, then the positive equilibrium, at which a constant number of adult insects emerge every year, is globally asymptotically stable. These results are consistent with Bulmer's conclusion that periodical behaviour results if competition is more severe between age-classes than within age-classes.

In order to show the dynamical consistency mentioned above, we have assumed in Theorem 2.2 that the competition intensity between age-classes depends only on their unidirectional age-distance. This assumption fails if there are some age-specific density effects. Such could take place when predators attack only adult individuals as observed in periodical cicadas. Therefore, our results seem not applicable directly to the case of periodical cicadas, which are one of the most famous examples of periodical insects ; see, e.g. [14, 15, 24]. However, the assumption might be fulfilled for other periodical insects such as May beetles and the northern oak eggar since the possibility that intraspecific competition is a dominant factor maintaining their periodical behaviour is not denied [4].

As said above, we have imposed some strong assumption on the survival probabilities of our models. However, as far as we know, it is the first result on the global stability of nonlinear semelparous Leslie matrix models. Although a recent study by Diekamann and Planqué [14] shows a class of semelparous Leslie matrix models that periodical behaviour results for a large set of initial conditions, the possibility of bistability is not excluded in their models.

Acknowledgments

Y. Chow was partially supported by MOST Grant Number 108-2115-M-008-016, Taiwan and R. Kon by JSPS KAKENHI Grant Number 16K05279 and 20K03735, Japan. R. Kon is indebted to the Institute of Mathematics, Academia Sinica, Taipei, Taiwan for its hospitality during his stay in 2018.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Y. Chow was partially supported by Ministry of Science and Technology (MOST), Taiwan [grant number 108-2115-M-008-016] and R. Kon by Japan Society for the Promotion of Science (JSPS) KAKENHI [grant number 16K05279 and 20K03735].