![MathJax Logo](/templates/jsp/_style2/_tandf/pb2/images/math-jax.gif)
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.
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 [Citation4] studied a special case of the following system of difference equations:
(1)
(1) where
. 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
,
, denotes the number of age-i individuals at time t. For
, the product of the constant
and the function
represents the probability that an age-i individual survives a unit of time. The product of the constant
and the function
represents the fertility of an age-n individual. Here, we set all
. Thus, system (Equation1
(1)
(1) ) assumes that only the last age-class is reproductive. That is only
represents the number of adult individuals. In this sense, system (Equation1
(1)
(1) ) describes the population dynamics of semelparous species such as cicadas and beetles.
With in system (Equation1
(1)
(1) ), Bulmer [Citation4] 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 (Equation1
(1)
(1) ) has been studied in many papers. For example, Cushing and Li [Citation11] studied bifurcations that occur around the extinction equilibrium of (Equation1
(1)
(1) ) with n = 2 and classified the stability of bifurcating positive equilibria and 2-cycles. This study was extended by Cushing [Citation9] to the case n = 3; see also [Citation8, Citation10, Citation19]. 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. [Citation13] examined the asymptotic behaviour of (Equation1
(1)
(1) ) mathematically and numerically for the special case n = 2 and
, 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. [Citation25], Kon [Citation18], Kon and Iwasa [Citation20] and Diekmann and Planqué [Citation14]. All these studies only reveal the local behaviour around equilibria, cycles or the coordinate axes (but see [Citation14] for an example of (Equation1
(1)
(1) ) where the coordinate axes attract a large set of initial conditions).
This paper aims at the global behaviour of system (Equation1(1)
(1) ). We first try to find a certain class of (Equation1
(1)
(1) ) that is dynamically consistent with the following system of difference equations:
(2)
(2) That is
for
with
(3)
(3) Here,
and
are, respectively, the population size and the growth rate of species i at time t.
System (Equation2(2)
(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, [Citation2, Citation3, Citation7, Citation17, Citation23, Citation31]. Some specific examples of system (Equation2
(2)
(2) ) with n = 2 and 3 are reported in [Citation16, Citation26–29]. In Theorem 2.2, it is shown that under some conditions on system (Equation1
(1)
(1) ), there exists a sequence of non-singular matrices
with period n such that any solutions
of (Equation1
(1)
(1) ) and
of (Equation2
(2)
(2) ) satisfy
(4)
(4) This means systems (Equation1
(1)
(1) ) and (Equation2
(2)
(2) ) are dynamically consistent.
The next step is to determine the global dynamics of (Equation2(2)
(2) ). For a general system, this is not an easy task. Inspired by the Leslie–Gower competition model [Citation22], we are led to some simplified versions of (Equation2
(2)
(2) ) like the following
(5)
(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
with
. Furthermore,
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 (Equation5
(5)
(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 (Equation4(4)
(4) ), we finally get in Theorem 5.1 the global dynamics of semelparous Leslie matrix models (Equation1
(1)
(1) ) in which
(6)
(6) Because of the periodicity of
in (Equation4
(4)
(4) ), we will find that the set of single-species equilibria of (Equation5
(5)
(5) ) corresponds to a single-class n-cycle in system (Equation1
(1)
(1) ). Thus, we obtain that depending on whether c<1 or not, system (Equation1
(1)
(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 (Equation6
(6)
(6) ),
is a positive eigenvector of the linearized system of (Equation1
(1)
(1) ) at the origin and
the corresponding eigenvalue which is greater than 1 as
by assumption.
This paper is organized as follows. In Section 2, we derive a condition under which (Equation4(4)
(4) ) holds, thus determining a certain class of semelparous Leslie matrix models (Equation1
(1)
(1) ) which is dynamically consistent with the Kolmogorov difference equations (Equation2
(2)
(2) ). In Sections 3 and 4, we consider a special class of Kolmogorov difference equations that include (Equation5
(5)
(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 (Equation6
(6)
(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 (Equation1(1)
(1) ) is dynamically consistent with the Kolmogorov difference Equation (Equation2
(2)
(2) ). Note that system (Equation2
(2)
(2) ) has cyclic symmetry in the sense that if
is a solution of (Equation2
(2)
(2) ), then so is
. In fact, we have
where the subscripts of
are counted mod n and
denotes the ith component of the vector
.
Let and
. Assume that
(H1) |
|
It is clear that both and
are forward invariant under (Equation1
(1)
(1) ). If each
is differentiable, then the linearization of system (Equation1
(1)
(1) ) at the origin yields the linear difference equation
, where
Being a non-negative irreducible matrix, Perron–Frobenius Theorem ensures that U has a dominant eigenvalue
and a positive eigenvector, say,
associated with
. It is straightforward to show that
, where
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
satisfy
(7)
(7) which implies that once
is fixed,
are uniquely determined by
Because U is the Jacobian matrix of system (Equation1
(1)
(1) ) evaluated at the origin, the origin of (Equation1
(1)
(1) ) is asymptotically stable if
and unstable if
. Moreover,
implies that the vector
gives a stationary age-distribution for the linearized system
. In fact, if the initial population
is proportional to
, then so is
for each
. This motivates us to consider the following normalized population:
(8)
(8) where D is the diagonal matrix whose diagonal entries are
. We will see below how the desired dynamical consistency is obtained via
. Using (Equation7
(7)
(7) ) and (Equation8
(8)
(8) ), it is easy to check that (Equation1
(1)
(1) ) becomes
(9)
(9) We introduce the following assumption on the survival probabilities above:
(H2) |
|
Here, matrices P and D are given in (Equation3(3)
(3) ) and (Equation8
(8)
(8) ), respectively. Under this assumption, (Equation9
(9)
(9) ) is equivalent to
(10)
(10) which would be the same as system (Equation2
(2)
(2) ) if the subscripts of all y terms on the right-hand side are shifted forward by one.
Denote by and
the maps defined by (Equation2
(2)
(2) ) and (Equation9
(9)
(9) ), respectively. Similarly, the t-fold compositions of
and
with themselves are denoted by
and
, respectively. By definition, both
and
mean the identity map.
Lemma 2.1
Assume that (H1) and (H2) hold. Then and
are maps from
to itself and
(11)
(11) In particular,
for all
and
.
Proof.
Clearly, and
are maps from
to itself. Since
and
are the identity map, (Equation11
(11)
(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 (Equation2
(2)
(2) ) implies that
(12)
(12) Then assumption (H2) ensures that for all
,
(13)
(13) That verifies (Equation11
(11)
(11) ) for t = 1. Suppose
holds for some
. Then
where (Equation13
(13)
(13) ) was used for the last equality above. Applying (Equation12
(12)
(12) ) repeatedly, we obtain
and thus
This completes the proof of (Equation11
(11)
(11) ) by mathematical induction. The final assertion follows from the fact that
is the identity matrix for any
.
The desired dynamical consistency relation (Equation4(4)
(4) ) with
follows from Lemma 2.1.
Theorem 2.2
Assume (H1) and (H2) hold. Let and
be solutions of (Equation1
(1)
(1) ) and (Equation2
(2)
(2) ), respectively. Then
for all
whenever
. Here, matrices P and D are given in (Equation3
(3)
(3) ) and (Equation8
(8)
(8) ), respectively.
Proof.
By (Equation8(8)
(8) ), the assumption
implies
. Thus, by Lemma 2.1,
holds for all
. This implies that
for all
. By (Equation8
(8)
(8) ), we finally obtain that
for all
.
3. Global dynamics of Kolmogorov difference equations
As shown in [Citation16], system (Equation2(2)
(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 (Equation2
(2)
(2) ): For
and
,
(14)
(14) where all
. Equation (Equation14
(14)
(14) ) means that the effect of the other species on the growth rate of species i is determined by their total population size
. We study system (Equation14
(14)
(14) ) under the assumptions that for
,
(A1) |
| ||||
(A2) | there are positive constants | ||||
(A3) |
|
(A1)′ |
| ||||
(A2)′ |
| ||||
(A3)′ |
|
It is straightforward to show that the functions
(16)
(16) satisfy (A1)–(A3) if all
and
are positive constants with
and
. Thus, the following system is an example of (Equation14
(14)
(14) ) satisfying (A1)–(A3): For
and
,
(17)
(17) When m = 1,
,
, and
, system (Equation17
(17)
(17) ) becomes
(18)
(18) which satisfies (A1)
–(A3)
and is a special case of the Leslie–Gower model [Citation22]:
(19)
(19) Here, we may assume without loss of generality that the carrying capacities
are positive and strictly decreasing in i. For n = 2, Cushing et al. [Citation7] 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 [Citation23] and Smith [Citation31]. For
, there are only some partial results. For instance, Ruiz-Herrera [Citation30], Chow and Hsieh [Citation5] and Ackleh et al. [Citation1] 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. [Citation3] gave a general result on higher dimensional monotone maps that guarantees the global asymptotic stability of an interior equilibrium of system (Equation19
(19)
(19) ) with n = 3 and all
. Chow and Palmer [Citation6] showed that when n = 3 and
, the unique interior equilibrium of system (Equation19
(19)
(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 (Equation14
(14)
(14) ) and (Equation18
(18)
(18) ) such that this conjecture holds.
It is clear that assumption (A1) implies both and
are forward invariant for system (Equation14
(14)
(14) ). The following lemma shows that any solution of system (Equation14
(14)
(14) ) converges to neither
nor infinity under assumption (A3).
Lemma 3.1
Assume that (A1) and (A3) hold. Then for any , the omega-limit set
of system (Equation14
(14)
(14) ) is a compact subset of
.
Proof.
Let ,
and
be a solution of (Equation14
(14)
(14) ). Remember L and
are defined in (Equation15
(15)
(15) ). Suppose that
. Then
holds for all
. By (Equation15
(15)
(15) ) and (Equation14
(14)
(14) ), we have
Thus, the solution
eventually enters
. Since the right-hand side of (Equation14
(14)
(14) ) is a continuous function of
and
is compact, the solution
is bounded. Thus,
is compact. Similarly, the first inequality in (A3) implies
with
eventually enters the compact set
with sufficiently small
. Since the right-hand side of (Equation14
(14)
(14) ) is a continuous function of
and all
by (A1), we can conclude that
.
Based on assumptions (A1) and (A2), we first show a preliminary lemma.
Lemma 3.2
Assume (A1) and (A2) hold. Then for any and
, both
and
are increasing for
.
Proof.
Let be given in (A2). Then
is positive, decreasing in x as
By (A2) and (A1),
increases and
decreases in x. Hence,
is increasing in x. Using
, the first claim follows from
Similarly,
increases in x by (A2). The remaining claim follows from multiplying these functions together and using
again.
Define and
. Using Lemmas 3.1 and 3.2, we now show the following result on global dynamics of system (Equation14
(14)
(14) ).
Theorem 3.3
Assume that (A1)–(A3) hold and is a solution of system (Equation14
(14)
(14) ) with
. Let
and
.
If all
and
, then
, where η is uniquely determined by
.
If all
and
, then
, where η is uniquely determined by
and
is the jth unit vector. In particular,
for
.
If all
, then
, where η is uniquely determined by
.
Proof.
Note that the existence and uniqueness of η in (a) –(c) is due to assumptions (A1) and (A3). Define . By symmetry, we may assume
.
Part (a). We show by induction on that for
,
(20)
(20) Assume that all
hold for some
. Note that all
by assumption. Then certainly
. Using (Equation14
(14)
(14) ) and the first claim in Lemma 3.2, we obtain
as follows:
where in the last inequality (A1) and
are used.
Since all ,
can be verified as follows:
(21)
(21) Using (Equation14
(14)
(14) ), (Equation21
(21)
(21) ) and (A1) again, we obtain
(22)
(22) Thus
and (Equation20
(20)
(20) ) is verified by induction.
Equation (Equation20(20)
(20) ) and Lemma 3.1 imply that
, where
so that (Equation20
(20)
(20) ) is valid for i = 1 as well. On the half-line Y, system (Equation14
(14)
(14) ) is reduced to n one-dimensional equations:
(23)
(23) Note that, by Lemma 3.2 and (A1), every solution on Y converges to the point
, where
is uniquely determined by
. Since
is compact by Lemma 3.1,
is bounded for
. The boundedness ensures that
is also nonempty and invariantly connected (e.g. see Theorem 5.2, LaSalle [Citation21]). Since
is a unique nonempty invariantly connected set in Y, we can conclude that
, i.e. for
,
(24)
(24) We will see below that
and thus
are independent of the initial condition.
Write . Then
by (Equation20
(20)
(20) ). Remember
. The defining equations for
and
above can be rewritten as
(25)
(25) Since all
by assumption and
by (Equation20
(20)
(20) ), we find that
(26)
(26) Because each
is strictly decreasing by (A1), equality in (Equation26
(26)
(26) ) holds by (Equation25
(25)
(25) ). Then all
and thus, all
as some
by assumption. We conclude from (Equation24
(24)
(24) ) and (Equation25
(25)
(25) ) that
for
and η is uniquely determined by
. This verifies the assertion in (a).
Part (b). Since by assumption, we have
for
and
for
. As was done to show (Equation20
(20)
(20) ), we show by induction on t that for
,
(27)
(27) As a consequence, there exist constants
with
such that
(28)
(28) Assume that all
hold for some
with equality held for
. The inequality
can be verified exactly as in the formula before (Equation21
(21)
(21) ). Since
, the inequality in (Equation21
(21)
(21) ) is reversed with equality held for
. So is the inequality in (Equation22
(22)
(22) ). Hence,
with equality held for
. This proves (Equation27
(27)
(27) ) by induction and thus (Equation28
(28)
(28) ) is verified.
Following the same arguments in the proof of Part (a), we have . Let
. On the half-line above, system (Equation14
(14)
(14) ) is reduced to
and
for
as
is bounded by Lemma 3.1 and
for
. Similar to (Equation24
(24)
(24) ), we can get from Lemma 3.2 and (Equation28
(28)
(28) ) that
(29)
(29) Here,
for
,
for
and
for
. Moreover, the defining equations for
and
,
are given by
(30)
(30) We claim
. Then the conclusion in Part (b) follows from (Equation29
(29)
(29) ) with η uniquely determined by
as shown in the first equality in (Equation31
(31)
(31) ).
Suppose the contrary that . Using
for
and
for
, we may rewrite (Equation30
(30)
(30) ) as
(31)
(31) Since all
by assumption and
for
by (Equation28
(28)
(28) ),
(32)
(32) for all
. Because all
decreases strictly by (A1), the equality in (Equation32
(32)
(32) ) holds by (Equation31
(31)
(31) ). This leads to a contradiction as
by assumption and
for
.
Part (c). When all , all
equal
. The inequalities in both (Equation21
(21)
(21) ) and (Equation22
(22)
(22) ) become equalities. So we have
That is,
(33)
(33) Therefore, system (Equation14
(14)
(14) ) for i = 1 can be written as
(34)
(34) Then
with η uniquely determined by
. By (Equation33
(33)
(33) ),
for
as claimed. The proof is complete.
As a consequence, we have the following result for system (Equation5(5)
(5) ).
Corollary 3.4
Assume h satisfies (A1)-(A3)
and
is a solution of (Equation5
(5)
(5) ) with
. Let
and
.
For
,
with η uniquely determined by
.
For c>1,
with η uniquely determined by
. In particular,
for
.
For c = 1,
with η uniquely determined by
.
We remark that since implies
for all
, the results above can be easily extended to
. Depending on c<1, c = 1 or c>1, the asymptotic behaviour of system (Equation5
(5)
(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 (Equation14(14)
(14) ). For this purpose, we have to assume that besides (A1)–(A3), all
in system (Equation14
(14)
(14) ) are differentiable. Note that
by assumption (A1). Theorem 3.3 (a) says that
is the unique interior equilibrium with
uniquely determined by
(35)
(35) Furthermore, Theorem 3.3(b) says that if
, then
and the solution
of (Equation14
(14)
(14) ) converges to the single-species equilibrium
, where η is uniquely determined by
(36)
(36) Concerning the local asymptotic stability of these equilibria, we have the following result.
Theorem 4.1
Besides (A1) –(A3), we assume all on
.
If all
and
, then
is locally asymptotically stable.
If all
and
, then each
,
, is locally asymptotically stable.
Proof.
Part (a). Let be the Jacobian matrix of system (Equation14
(14)
(14) ) evaluated at
. Using (Equation35
(35)
(35) ),
is an
matrix whose diagonal and off-diagonal entries are, respectively, given by
where
are defined in (Equation35
(35)
(35) ). Being a circulant matrix [Citation12], its eigenvalues are
for
. Using
and all
, we get
(37)
(37) Because all
with some
by assumption and
for
,
(38)
(38) We claim that for
,
(39)
(39) Together with (Equation37
(37)
(37) ) and (Equation38
(38)
(38) ), we then have all
and thus
is asymptotically stable. Using
, the first inequality above follows easily as:
It remains to show
. By assumption (A2),
which yields
. Applying this inequality to (Equation37
(37)
(37) ), we get
as
by (Equation35
(35)
(35) ) and
by (A3).
Part (b). By symmetry, it suffices to consider . It is straightforward to show that
satisfies
for
. Moreover, (Equation36
(36)
(36) ) implies that
(40)
(40) Being an upper triangular matrix,
are eigenvalues of
. Because all
with some
and
decreases strictly by (A1), we get from (Equation36
(36)
(36) ) that
(41)
(41) By Lemma 3.2,
is increasing. Therefore,
(42)
(42) where we have used (Equation36
(36)
(36) ) and the assumption that all
,
. Combining together (Equation40
(40)
(40) )–(Equation42
(42)
(42) ), we obtain that all eigenvalues
lie in
. The assertion that
is asymptotically stable is verified.
Note that functions defined in (Equation16
(16)
(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
, the interior equilibrium
of system (Equation14
(14)
(14) ) is globally asymptotically stable in
if all
and
. Furthermore, every single-species equilibrium
,
, is globally asymptotically stable in
if all
and
.
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 (Equation1(1)
(1) ). We need to impose some extra conditions on the growth rate function g in system (Equation2
(2)
(2) ):
(H3) |
|
Then, Corollary 3.4 can be applied to system (Equation2(2)
(2) ) and global dynamics of system (Equation1
(1)
(1) ) follows immediately from Theorem 2.2.
Before stating the results, we note that by (H1) and
by (A3)
. Then (H2) and (H3) imply that
(43)
(43) where function h satisfies (A1)
–(A3)
and
. For
, the constant c above measures the effect that the normalized density
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
. For example, if
and g are given by
then (H2) and (H3) are satisfied with
. With
, we may use (Equation43
(43)
(43) ) to rewrite system (Equation1
(1)
(1) ) as
(44)
(44) Remember that matrices P and D are defined in (Equation3
(3)
(3) ) and (Equation8
(8)
(8) ), respectively.
Theorem 5.1
Assume for
and h satisfies (A1)
–(A3)
with
. Let
be a solution of (Equation44
(44)
(44) ) with
. Define
and
.
If
, then
, where
with η uniquely determined by
.
If c>1, then
converges to a cycle as
, i.e.
, where
for
and η is uniquely determined by
. In particular,
converges to the n-cycle
as
if
, i.e.
, where
.
If c = 1, then
converges to a cycle as
, i.e.
, where
for
and η is uniquely determined by
.
If h is also differentiable, then is globally asymptotically stable in
if
and
is globally asymptotically stable in
if c>1.
Proof.
Let and
be the solution of system (Equation5
(5)
(5) ) with the initial vector
. Note that
is positive.
Part (a). By Corollary 3.4, converges to
as
. Then, Theorem 2.2 implies that
as
. Since it holds for any positive
, we conclude that
as
.
Part (b). Note that and
. As above, we get from Corollary 3.4 that
as
. By Theorem 2.2,
as
for every
. Thus
converges to the cycle
.
Part (c). By Corollary 3.4, converges to
as
. Using Theorem 2.2,
as
for every
. Thus
converges to the cycle
.
The assertion on the asymptotic stability follows from Theorem 4.1.
Note that the set has measure zero as it is a subset of
, where
Therefore, we find that generic solutions of (Equation1
(1)
(1) ) converge to the n-cycle
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. [Citation14, Citation15, Citation24]. 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 [Citation4].
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 [Citation14] 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
References
- A.S. Ackleh, R.J. Sacker, and P. Salceanu, On a discrete selection-mutation model, J. Differ. Equ. Appl. 20 (2017), pp. 1383–1403. doi: 10.1080/10236198.2014.933819
- S. Baigent and Z. Hou, Global stability of discrete-time competitive population models, J. Differ. Equ. Appl. 23 (2017), pp. 1378–1396. doi: 10.1080/10236198.2017.1333116
- E.C. Balreira, S. Elaydi, and R. Luıs, Global stability of higher dimensional monotone maps, J. Differ. Equ. Appl. 23 (2017), pp. 2037–2071. doi: 10.1080/10236198.2017.1388375
- M.G. Bulmer, Periodical insects, Amer. Natur. 111 (1977), pp. 1099–1117. doi: 10.1086/283240
- Y. Chow and J. Hsieh, On multidimensional discrete-time Beverton–Holt competition models, J. Differ. Equ. Appl. 19 (2013), pp. 491–506. doi: 10.1080/10236198.2012.656618
- Y. Chow and K. Palmer, On a discrete three-dimensional Leslie–Gower competition model, Discrete Contin. Dyn. Syst. Ser. B 24 (2019), pp. 4367–4377.
- J. Cushing, S. Levarge, N. Chitnis, and S.M. Henson, Some discrete competition models and the competitive exclusion principle, J. Differ. Equ. Appl. 10 (2004), pp. 1139–1151. doi: 10.1080/10236190410001652739
- J.M. Cushing, Nonlinear semelparous Leslie models, Math. Biosci. Eng. 3 (2006), pp. 17–36. doi: 10.3934/mbe.2006.3.17
- J.M. Cushing, Three stage semelparous Leslie models., J. Math. Biol. 59 (2009), pp. 75–104. doi: 10.1007/s00285-008-0208-9
- J.M. Cushing and S.M. Henson, Stable bifurcations in semelparous Leslie models, J. Biol. Dyn. 6 (2012), pp. 80–102. doi: 10.1080/17513758.2012.716085
- J.M. Cushing and J. Li, On Ebenman's model for the dynamics of a population with competing juveniles and adults, Bull. Math. Biol. 51 (1989), pp. 687–713. doi: 10.1016/S0092-8240(89)80058-8
- P. Davis, Circulant Matrices, Wiley-Interscience, New York, NY, 1979.
- N.V. Davydova, O. Diekmann, and S.A. van Gils, Year class coexistence or competitive exclusion for strict biennials?, J. Math. Biol. 46 (2003), pp. 95–131. doi: 10.1007/s00285-002-0167-5
- O. Diekmann and R. Planqué, The winner takes it all: how semelparous insects can become periodical, J. Math. Biol. (2019). Available at https://doi.org/10.1007/s00285-019-01362-3.
- F.C. Hoppensteadt and J.B. Keller, Synchronization of periodical cicada emergences, Science 194 (1976), pp. 335–337. doi: 10.1126/science.987617
- H. Jiang and T.D. Rogers, The discrete dynamics of symmetric competition in the plane, J. Math. Biol. 25 (1987), pp. 573–596. doi: 10.1007/BF00275495
- J. Jiang and L. Niu, On the equivalent classification of three-dimensional competitive Leslie/Gower models via the boundary dynamics on the carrying simplex, J. Math. Biol. 74 (2017), pp. 1223–1261. doi: 10.1007/s00285-016-1052-y
- R. Kon, Competitive exclusion between year-classes in a semelparous biennial population, in Mathematical Modeling of Biological Systems, Vol. II, A. Deutsch, R.B. de la Parra, R.J. de Boer, O. Diekmann, P. Jagers, E. Kisdi, M. Kretzschmar, P. Lansky, and H. Metz, eds., Birkhäuser, Boston, MA, 2007, pp. 79–90.
- R. Kon, Stable bifurcations in multi-species semelparous population models, in Advances in Difference Equations and Discrete Dynamical Systems, Springer Proceedings in Mathematics & Statistics, 212, S. Elaydi, Y. Hamaya, H. Matsunaga and C. Pötzsche, eds., Springer, Singapore, 2017, pp. 3–25.
- R. Kon and Y. Iwasa, Single-class orbits in nonlinear Leslie matrix models for semelparous populations, J. Math. Biol. 55 (2007), pp. 781–802. doi: 10.1007/s00285-007-0111-9
- J.P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, PA, 1976.
- P.H. Leslie and J.C. Gower, The properties of a stochastic model for two competing species, Biometrika 45 (1958), pp. 316–330. doi: 10.1093/biomet/45.3-4.316
- P. Liu and S.N. Elaydi, Discrete competitive and cooperative models of Lotka–Volterra type, J. Comput. Anal. Appl. 3 (2001), pp. 53–73.
- J. Machta, J.C. Blackwood, A. Noble, A.M. Liebhold, and A. Hastings, A hybrid model for the population dynamics of periodical cicadas, Bull. Math. Biol. 81 (2018), pp. 1122–1142. doi: 10.1007/s11538-018-00554-0
- E. Mjølhus, A. Wikan, and T. Solberg, On synchronization in semelparous populations, J. Math. Biol.50 (2005), pp. 1–21. doi: 10.1007/s00285-004-0275-5
- L.I.W. Roeger, Hopf bifurcations in discrete May–Leonard competition models, Can. Appl. Math. Q.11 (2003), pp. 175–194.
- L.I.W. Roeger, Discrete May–Leonard competition models III, J. Difference Equ. Appl. 10 (2004), pp. 773–790. doi: 10.1080/10236190410001647825
- L.I.W. Roeger, Discrete May–Leonard competition models II, Discrete Contin. Dyn. Syst. Ser. B 5 (2005), pp. 841–860.
- L.I.W. Roeger and L.J. Allen, Discrete May–Leonard competition models I, J. Differ. Equ. Appl. 10 (2004), pp. 77–98. doi: 10.1080/10236190310001603662
- A. Ruiz-Herrera, Exclusion and dominance in discrete population models via the carrying simplex, J. Differ. Equ. Appl. 19 (2013), pp. 96–113. doi: 10.1080/10236198.2011.628663
- H.L. Smith, Planar competitive and cooperative difference equations, J. Differ. Equ. Appl. 3 (1998), pp. 335–357. doi: 10.1080/10236199708808108