On existence and uniqueness of a modified carrying simplex for discrete Kolmogorov systems

ABSTRACT

For a $C^1$ map T from $C=[0,+\mathrm{\infty }{)}^N$ to C of the form $T_i\left(x\right)=x_i{f}_i\left(x\right)$, the dynamical system $x\left(n\right)=T^n\left(x\right)$ as a population model is competitive if $\frac{\mathrm{\partial }{f}_i}{\mathrm{\partial }{x}_j}\le 0$ $(i\ne j)$. A well know theorem for competitive systems, presented by Hirsch [On existence and uniqueness of the carrying simplex for competitive dynamical systems, J. Biol. Dyn. 2(2) (2008), pp. 169–179] and proved by Ruiz-Herrera [Exclusion and dominance in discrete population models via the carrying simplex, J. Differ. Equ. Appl. 19(1) (2013), pp. 96–113] with various versions by others, states that, under certain conditions, the system has a compact invariant surface $\mathrm{\Sigma }\subset C$ that is homeomorphic to ${\mathrm{\Delta }}^{N-1}=\{x\in C:x_1+\cdots +x_N=1\}$, attracting all the points of $C\setminus \left\{0\right\}$, and called carrying simplex. The theorem has been well accepted with a large number of citations. In this paper, we point out that one of its conditions requiring all the $N^2$ entries of the Jacobian matrix $Df=\left(\frac{\mathrm{\partial }{f}_i}{\mathrm{\partial }{x}_j}\right)$ to be negative is unnecessarily strong and too restrictive. We prove the existence and uniqueness of a modified carrying simplex by reducing that condition to requiring every entry of Df to be nonpositive and each $f_i$ is strictly decreasing in $x_i$. As an example of applications of the main result, sufficient conditions are provided for vanishing species and dominance of one species over others.

Keywords:

• Discrete competitive models
• retrotone maps
• carrying simplex
• existence and uniqueness
• dominant species
• vanishing species

2010 Mathematics Subject Classifications:

• 37B35
• 37D10
• 37N25
• 92D25

Acknowledgements

The author consulted Professor Stephen Baigent on this topic and is grateful for his encouragement of writing up this paper. The author is also grateful to the referees and editors for their comments and suggestions adopted in this version of the paper.

Disclosure statement

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