Does a population with the highest turnover coefficient win competition?

TL;DR

This paper analyzes a discrete time competition model with populations differing in fertility and mortality, comparing its dynamics to continuous models and identifying conditions for competitive exclusion and periodic solutions.

Contribution

It provides new sufficient conditions for competitive exclusion and periodic solutions in a discrete competition model, extending classical logistic, Beverton-Holt, and Ricker models.

Findings

Conditions for competitive exclusion established

Existence of periodic solutions demonstrated

Comparison with continuous models elucidated

Abstract

We consider a discrete time competition model. Populations compete for common limited resources but they have different fertilities and mortalities rates. We compare dynamical properties of this model with its continuous counterpart. We give sufficient conditions for competitive exclusion and the existence of periodic solutions related to the classical logistic, Beverton-Holt and Ricker models.

Equations

$$x_i(t+1)=x_i(t)-d_i{x}_i(t)+b_i{x}_i(t)f(\mathbf{x}(t)).$$

$$f(\mathbf{x})=\{\begin{array}{ll}1,& \text{if (1+b1)x1+⋯+(1+bk)xk≤K},\\ \frac{K-x_1-\cdots -x_k}{b_1{x}_1+\cdots +b_k{x}_k},& \text{if (1+b1)x1+⋯+(1+bk)xk>K}.\end{array}$$

$$x_i^{\prime }(t)=-d_i{x}_i(t)+b_i{x}_i(t)f(\mathbf{x}(t)),i=1,\dots ,k.$$

$$f(\mathbf{x})\leq\min\bigg{\{}\frac{d_{i}}{b_{i}}\colon\,\,i=1,\dots,k\bigg{\}}\quad\textrm{if $|\mathbf{x}|\geq M$},$$

$$L_1>L_2\ge \cdots \ge L_k.$$

$$\underset{t\to \infty }{\lim }{x}_i(t)=0\text{ for i≥2}.$$

$$(b_i^{-1}\ln x_i(t){)}^{\prime }=-L_i^{-1}+f(\mathbf{x}(t)).$$

$$(b_1^{-1}\ln x_1(t)-b_i^{-1}\ln x_i(t){)}^{\prime }=L_i^{-1}-L_1^{-1}.$$

$$\frac{d}{dt}\ln\bigg{(}\frac{x_{1}(t)^{b_{i}}}{x_{i}(t)^{b_{1}}}\bigg{)}=b_{1}b_{2}(L_{i}^{-1}-L_{1}^{-1})>0$$

$$\underset{t\to \infty }{\lim }\frac{x_1(t{)}^{b_i}}{x_i(t{)}^{b_1}}=\infty .$$

$$\frac{x_i(t+1)-x_i(t)}{b_i{x}_i(t)}=-L_i^{-1}+f(\mathbf{x}(t)).$$

$$\frac{1}{b_1}{D}_l{x}_1(t)-\frac{1}{b_i}{D}_l{x}_i(t)=L_i^{-1}-L_1^{-1}.$$

$$D_l{x}_i(t):=\frac{x_i(t+1)-x_i(t)}{x_i(t)}.$$

$$D_l{x}_1(t)=\alpha D_l{x}_i(t)+\beta .$$

$$\frac{u_{n+1}-u_n}{u_n}=\alpha \frac{v_{n+1}-v_n}{v_n}+\beta ,$$

$$g_n(x_1,\dots ,x_n)=(\alpha x_1+\gamma )(\alpha x_2+\gamma )\cdots (\alpha x_n+\gamma )$$

$$g_n(x_1,\dots ,x_n)\ge {\left(\alpha m^{1/n}+\gamma \right)}^n.$$

$$L(x_1,\dots ,x_n)=g_n(x_1,\dots ,x_n)+\lambda (m-x_1{x}_2\cdots x_n).$$

$$\frac{\partial L(x_1,\dots ,x_n)}{\partial x_i}=\frac{\alpha g_n(x_1,\dots ,x_n)}{\alpha x_i+\gamma }-\frac{\lambda x_1{x}_2\cdots x_n}{x_i}.$$

$$\frac{u_{n+1}}{u_n}=\alpha \frac{v_{n+1}}{v_n}+\gamma ,$$

$$u_{n_i}=u_0{g}_{n_i}(x_1,\dots ,x_{n_i})\ge u_0{\left(\alpha m^{1/n_i}+\gamma \right)}^{n_i}\ge u_0{\left(\alpha {\overline{m}}^{1/n_i}+\gamma \right)}^{n_i}.$$

$$(\alpha \theta +\gamma )(\alpha {\theta }^{-1}+\gamma )=1.$$

$$\theta +{\theta }^{-1}=L,\text{ where L=α∣γ∣α2+γ2−1}.$$

$$\frac{u_{2n}}{u_{2n-1}}=\alpha \theta +\gamma =\alpha \frac{v_{2n}}{v_{2n-1}}+\gamma ,$$

$$\frac{u_{2n+1}}{u_{2n}}=\frac{1}{\alpha \theta +\gamma }=\alpha {\theta }^{-1}+\gamma =\alpha \frac{v_{2n+1}}{v_{2n}}+\gamma .\text{qed}$$

$$X=\{\mathbf{x}\in {\mathbb{R}}_{+}^k:x_1+\cdots +x_k\le \overline{M}\}$$

$$T_i(\mathbf{x})\ge (1+\delta )x_i\text{for i=1,…,k and x∈B(0,ε)},$$

$$b_1(1-d_i)\le b_i(1-d_1)\text{for i=2,…,k.}$$

$$y(t+1)=S(y(t)),$$

$$\underset{t\to \infty }{\lim }{x}_i(t)=0\text{ for i=2,…,k}.$$

Full text

1. Introduction

It is well known that if species compete for common limited resources, then usually they cannot coexist in the long term. This law was introduced by Gause [7] and it is called the principle of competitive exclusion. There are a lot of papers where the problem of competitive exclusion or coexistence is discussed. Most of them are described by continuous time models, but there are also some number of discrete time models devoted to this subject (see [1, 6] and the references given there). If we have continuous and discrete time versions of a similar competition model it is interesting to compare the properties of both versions of the model, especially to check if they are dynamically consistent, i.e., if they possess the same dynamical properties as stability or chaos. In this paper we consider a discrete time competition model with overlapping generations. We prove a sufficient condition for competitive exclusion and compare it with its continuous counterpart.

The model considered here is the following. A population consists of $k$ different individual strategies. The state of the population at time $t$ is given by the vector $\mathbf{x}(t)=[x_{1}(t),\dots,x_{k}(t)]$, where $x_{i}(t)$ is the size of subpopulation with the phenotype $i$. Individuals with different phenotypes do not mate and each phenotype $i$ is characterized by per capita reproduction $b_{i}$ and mortality $d_{i}$. We assume that the juvenile survival rate depends on the state $\mathbf{x}$ and it is given by a function $f\colon\mathbb{R}^{k}_{+}\to[0,1]$. Therefore, $f$ describes the suppression of growth driven, for example, by competition for food or free nest sites for newborns. The time evolution of the state of population is given by the system

[TABLE]

We assume that $0<d_{i}\leq 1$, $d_{i}<b_{i}$ for $i=1,\dots,k$ and $f(\mathbf{x})>0$ for $\mathbf{x}\neq 0$. The model is similar in spirit to that presented in [3] (a continuous version) and in [4] (a discrete version) but in those papers $f$ has a special form strictly connected with competition for free nest sites for newborns. A simplified Leslie/Gower model [2] is also of the form  (1). The suppression function $f$ can be quite arbitrary. Usually, it is of the form $f(\mathbf{x})=\varphi(w_{1}x_{1}+\dots+w_{k}x_{k})$, where $\varphi$ is a decreasing function and $w_{1},\dots,w_{k}$ are positive numbers, but e.g. in [3] it is of the form

[TABLE]

Now we present some motivation for studying model (1). We begin with a continuous time version of it. The time evolution of the state of population is described by the system

[TABLE]

We assume that $0<d_{i}<b_{i}$, $f$ has values in the interval $[0,1]$, and

[TABLE]

where $|\mathbf{x}|=x_{1}+\dots+x_{k}$. From the last condition it follows that the total size $|\mathbf{x}(t)|$ of the population is bounded by $\max(M,|\mathbf{x}(0)|)$. We also assume that $f$ is a sufficiently smooth function to have the existence and uniqueness of the solutions of (2), for example it is enough to assume that $f$ satisfies a local Lipschitz condition.

We denote by $L_{i}=b_{i}/d_{i}$ the turnover coefficient for the strategy $i$. We assume that

[TABLE]

It is well known that

[TABLE]

Indeed, from (2) it follows that

[TABLE]

Thus

[TABLE]

Therefore

[TABLE]

and, consequently,

[TABLE]

Since $x_{1}(t)$ is a bounded function, from (9) it follows that (5) holds.

Now we return to a discrete time version of model (2). From (3) it follows immediately that $x_{i}(t+1)\leq x_{i}(t)$ if $|\mathbf{x}(t)|\geq M$ and, therefore, the sequence $(x_{i}(t))$ is bounded. Moreover, since $d_{i}\leq 1$ we have $x_{i}(t)>0$ if $x_{i}(0)>0$. It is of interest to know whether (5) holds also for discrete model (1). Observe that (1) can be written as

[TABLE]

Now (7) takes the form

[TABLE]

In the last formula the logarithmic derivative $x_{i}^{\prime}/x_{i}$ was replaced by its discrete version

[TABLE]

Let $\alpha=b_{1}/b_{i}$ and $\beta=b_{1}(L_{i}^{-1}-L_{1}^{-1})=\alpha d_{i}-d_{1}$. Then $0<\beta<\alpha$ and (11) can be written in the following way

[TABLE]

We want to find a sufficient condition for (5). In order to do it we formulate the following general question, which can be investigated independently of the above biological models.

Problem 1*.*

Find parameters $\alpha$ and $\beta$, $0<\beta<\alpha$, such that the following condition holds:

(C)  if $(u_{n})$ and $(v_{n})$ are arbitrary bounded sequences of positive numbers satisfying

[TABLE]

for $n\in\mathbb{N}$, then $\lim\limits_{n\to\infty}v_{n}=0$.

In the case when the model has the property of competitive exclusion (5) one can ask if the dynamics of the $k$-dimensional model is the same as in the restriction to the one-dimensional model. The answer to this question is positive for the continuous version, because the one-dimensional model has a very simple dynamics. In Section 3.2 we also show that both dynamics are similar if the one-dimensional model has the shadowing property. More interesting question is what can happen when condition (C) does not hold. One can expect that then subpopulations with different strategies can coexist even if condition (4) holds. But we do not have a coexistence equilibrium (i.e. a positive stationary solution of (1)) which makes the problem more difficult. In Section 3.3 we check that two-dimensional systems with $f$ related to the classical logistic, Beverton-Holt and Ricker models can have periodic solutions even in the case when one-dimensional versions of these models have stationary globally stable solutions and the two-dimensional model has a locally stable boundary equilibrium $(x_{1}^{*},0)$.

2. Competitive exclusion

The solution of Problem 1 is formulated in the following theorem.

Theorem 1**.**

If  $\alpha\leq 1+\beta$ then condition (C) is fulfilled. If $\alpha>1+\beta$ then we can find periodic sequences $(u_{n})$ and $(v_{n})$ of period two with positive elements which satisfy $(\ref{p1})$.

Lemma 1**.**

Consider the function

[TABLE]

defined on the set $S_{n,m}=\{\mathbf{x}\in\mathbb{R}^{n}_{+}\colon\,x_{1}x_{2}\cdots x_{n}=m\}$, where $\alpha>0$, $\gamma\geq 0$, $m>0$ and $n$ is a positive integer. Then

[TABLE]

Proof.

The case $\gamma=0$ is obvious, so we assume that $\gamma>0$. We use the standard technique of the Lagrange multipliers for investigating problems on conditional extrema. Let

[TABLE]

Then

[TABLE]

Observe that if $\dfrac{\partial L(x_{1},\dots,x_{n})}{\partial x_{i}}=0$ for $i=1,\dots,n$, then $x_{1}=\dots=x_{n}=m^{1/n}$. It means that the function $g_{n}$ has only one local conditional extremal point and this point is the global minimum because $g_{n}(\mathbf{x})$ converges to infinity as $\|\mathbf{x}\|\to\infty$. ∎

Proof of Theorem 1.

Equation (13) can be written in the following form

[TABLE]

where $\gamma=\beta+1-\alpha$. Consider the case $\alpha\leq 1+\beta$. Then $\gamma\geq 0$ and $\alpha+\gamma=\beta+1>1$. We show that if $(v_{n})$ is a sequence of positive numbers such that $\limsup\limits_{n\to\infty}v_{n}>0$ and $(u_{n})$ is a sequence of positive numbers such that (16) holds, then the sequence $(u_{n})$ is unbounded. Indeed, then we can find an $\overline{m}>0$ and a subsequence $(v_{n_{i}})$ of $(v_{n})$ such that $v_{n_{i}}\geq\overline{m}$ for $i\in\mathbb{N}$. We set $v_{0}=1$ and $x_{i}=v_{i}/v_{i-1}$ for $i\in\mathbb{N}$. Then $v_{n}=x_{1}\cdots x_{n}$ and $u_{n}=u_{0}g_{n}(x_{1},\dots,x_{n})$, where $u_{0}=u_{1}/(\alpha v_{1}+\gamma)$. If $m=x_{1}\cdots x_{n_{i}}$, then $m\geq\overline{m}$ and from Lemma 1 it follows that

[TABLE]

Since $\lim\limits_{i\to\infty}\overline{m}^{1/n_{i}}=1$ and $\alpha+\gamma>1$ we obtain $\lim\limits_{i\to\infty}u_{n_{i}}=\infty$, which proves the first part of the theorem. Now we assume that $\alpha>1+\beta$. Then $\gamma<0$. First we check that there exists $\theta>1$ such that

[TABLE]

Equation (17) can be written in the following form

[TABLE]

Since $\alpha+\gamma=\beta+1>1$ we have $\alpha^{2}+\gamma^{2}-1>2\alpha|\gamma|$, which gives $L>2$ and implies that there exists $\theta>1$ such that (17) holds. Now we put $u_{2n-1}=c_{1}$, $u_{2n}=c_{1}(\alpha\theta+\gamma)$, $v_{2n-1}=c_{2}$, $v_{2n}=c_{2}\theta$ for $n\in\mathbb{N}$, where $c_{1}$ and $c_{2}$ are any positive constants. Then

[TABLE]

and using (17) we obtain

[TABLE]

Remark 1*.*

We have proved a slightly stronger condition than (C) in the case $\alpha\leq 1+\beta$. Namely, if $(u_{n})$ is a bounded sequence of positive numbers, $(v_{n})$ is a sequence of positive numbers and they satisfy (13), then $\lim\limits_{n\to\infty}v_{n}=0$. In the proof of condition (C) we have not used the preliminary assumption that $\beta<\alpha$.

3. Applications

Now we return to the model given by (1). We assume that $f\colon\mathbb{R}^{k}_{+}\to[0,1]$ is a continuous function which satisfies (3). From (3) it follows that there exists $\overline{M}>0$ such that the set

[TABLE]

is invariant under (1), i.e., if $\mathbf{x}(0)\in X$ then $\mathbf{x}(t)\in X$ for $t>0$. We restrict the domain of the model to the set $X$. Let $T\colon X\to X$ be the transformation given by $T_{i}(\mathbf{x})=(1-d_{i})x_{i}+b_{i}f(\mathbf{x})x_{i}$, for $i=1,\dots,k$.

3.1. Persistence

First we check that if $f(\mathbf{0})=1$ then the population is persistent, i.e., $\liminf_{n\to\infty}\|T^{n}(\mathbf{x})\|\geq\varepsilon_{1}>0$ for all $\mathbf{x}\neq\mathbf{0}$. This is a standard result from persistence theory but we check it to make the paper self-contained. Since $b_{i}>d_{i}$ for $i=1,\dots,k$ we find $\varepsilon>0$ and $\delta>0$ such that

[TABLE]

where $B(\mathbf{0},\varepsilon)$ denotes the open ball in $X$ with center $\mathbf{0}$ and radius $\varepsilon$. Moreover, since $T(\mathbf{x})\neq\mathbf{0}$ for $\mathbf{x}\neq 0$ the closed set $T(X\setminus B(\mathbf{0},\varepsilon))$ is disjoint with some neighbourhood of $\mathbf{0}$. Using (19) we find $\varepsilon_{1}\in(0,\varepsilon)$ such that for each $\mathbf{x}\neq\mathbf{0}$ we also find an integer $n_{0}(\mathbf{x})$ such that $T^{n}(\mathbf{x})\notin B(\mathbf{0},\varepsilon_{1})$ for $n\geq n_{0}(\mathbf{x})$.

3.2. Convergence to one-dimensional dynamics

Now we present some corollaries of Theorem 1 concerning the long-time behaviour of the population. The inequality $0<\alpha\leq 1+\beta$ can be written in terms of birth and death coefficients as

[TABLE]

It means that if (4) and (20) hold, then all strategies except the first one become extinct. It suggests that the model should behave asymptotically as $t\to\infty$, like a one-dimensional model corresponding to a population consisting of only the first strategy. This reduced model is given by the recurrent formula

[TABLE]

where $S(y)=y-d_{1}y+b_{1}yf(y,0,\dots,0)$.

In order to check that the model given by (1) has the same asymptotic behaviour as the transformation $S$, we need some auxiliary definitions. A sequence $(y_{k})$ is called an $\eta$-pseudo-orbit of a transformation $S$ if $|S(y_{k})-y_{k+1}|<\eta$ for all $k\geq 1$. The transformation $S$ is called shadowing, if for every $\delta>0$ there exists $\eta>0$ such that for each $\eta$-pseudo-orbit $(y_{k})$ of $S$ there is a point $y$ such that $|y_{k}-S^{k}(y)|<\delta$.

Theorem 2**.**

Assume that $f(\mathbf{0})=1$ and that conditions $(\ref{ineq})$, $(\ref{a:1})$ hold. Then

[TABLE]

If $S$ is shadowing then for each $\delta>0$ and for each initial point $\mathbf{x}(0)=(x_{1},\dots,x_{k})$ with $x_{1}>0$ there exists $t_{0}\geq 0$ such that $y(t_{0})>0$ and

[TABLE]

Proof.

Let us fix a $\delta>0$ and let $\eta>0$ be a constant from the shadowing property of $S$. From the uniform continuity of the function $f$ there is an $\varepsilon>0$ such that

[TABLE]

Since all strategies except the first one become extinct, there exists $t_{0}\geq 0$ such that $x_{2}(t)+\dots+x_{k}(t)<\varepsilon$ for $t\geq t_{0}$. From (23) it follows that

[TABLE]

and, consequently, the sequence $x_{1}(t_{0}),x_{1}(t_{0}+1),\dots$ is an $\eta$-pseudo-orbit. Since $S$ is shadowing we have (22). ∎

The shadowing property was intensively studied for the last thirty years and there are a lot of results concerning the shadowing property for one-dimensional maps (cf. a survey paper by Ombach and Mazur [12]). It is obvious that if $S$ has an asymptotically stable periodic orbit then $S$ is shadowing on the basin of attraction of this orbit. Moreover, for a continuous one-dimensional transformation the convergence of all iterates to a unique fixed point $x$ implies its global stability [15]. Thus, as a simple consequence of Theorem 2 we obtain

Corollary 1**.**

Assume that $f(\mathbf{0})=1$ and that conditions $(\ref{ineq})$, $(\ref{a:1})$ hold. If $S$ has a fixed point $x_{*}>0$ and $\lim\limits_{n\to\infty}S^{n}(x)=x_{*}$ for all $x>0$, then for each initial point $\mathbf{x}(0)=(x_{1},\dots,x_{k})$ with $x_{1}>0$, we have $\lim\limits_{t\to\infty}\mathbf{x}(t)=(x_{*},0,\dots,0)$.

Some applications of shadowing to semelparous population similar to Theorem 2 and Corollary 1 can be found in [14]. An interested reader will find there also some observations concerning chaotic behaviour of such models. In particular, the model given by (1) can exhibit chaotic behaviour if the suppression function is of the form $f(\mathbf{x})=1-x_{1}-\dots-x_{n}$, i.e., it is a generalization of the logistic model.

Remark 2* (Dynamical consistence).*

If we replace $x^{\prime}_{i}(t)$ with $(x_{i}(t+h)-x_{i}(t))/h$ in (2) then we get

[TABLE]

One can ask if this scheme is dynamically consistent with (2). Observe, that inequalities (4) also hold if we replace $b_{i}$ with $b_{i}h$ and $d_{i}$ with $d_{i}h$. The difference equation (24) is said to be dynamically consistent with (2) if they possesses the same dynamical behavior such as local stability, bifurcations, and chaos [9], or more specifically [11, 13] if they have the same given property, e.g. if the competitive exclusion takes place in both discrete and continuous models. The model (24) is biologically meaningful only if the death coefficients are $\leq 1$, i.e., if

[TABLE]

We assume that $b_{i}$ and $d_{i}$ satisfy (4), i.e., $b_{1}d_{i}>b_{i}d_{1}$ for $i=2,\dots,k$. Let $b_{i,h}=b_{i}h$, $d_{i,h}=d_{i}h$. Now, (20) applied to $b_{i,h}$ and $d_{i,h}$ gives

[TABLE]

In particular if (4) holds and $b_{i}\geq b_{1}$ for $i=2,\dots,k$ then for all $h$ satisfying (25) all strategies except the first one become extinct, i.e., the difference equation (24) is dynamically consistent with (2) with respect to this property. We cannot expect “full” dynamical consistence if the above conditions hold, because in the case of the logistic map, i.e., if $f(\mathbf{x})=1-x_{1}-x_{2}$, the stationary point $\mathbf{x}_{1}^{*}=((b_{1}-d_{1})d_{1}^{-1},0)$ of (2) is globally stable but in the numerical scheme (24) this point loses stability when $b_{1}h>2+d_{1}h$.

3.3. Periodic solutions

Theorem 1 can be also useful if we look for periodic oscillation in the model given by (1). We restrict our investigation to the two-dimensional model. We recall that if $\alpha>1+\beta$, then the periodic sequences given by $u_{2n-1}=c_{1}$, $u_{2n}=c_{1}(\alpha\theta+\gamma)$, $v_{2n-1}=c_{2}$, $v_{2n}=c_{2}\theta$ for $n\in\mathbb{N}$, satisfy $(\ref{p1})$. Here $c_{1}$ and $c_{2}$ are any positive constants, $\theta>1$ is a solution of the equation

[TABLE]

$\alpha=b_{1}/b_{2}$, $\beta=\alpha d_{2}-d_{1}>0$, and $\gamma=1+\beta-\alpha=\alpha(d_{2}-1)+(1-d_{1})<0$. Under these assumptions we are looking for $c_{1},c_{2}>0$ such that

[TABLE]

This system is equivalent to

[TABLE]

Since $f(\mathbf{x})\in(0,1)$ for $\mathbf{x}\in X\setminus\{\mathbf{0}\}$, we have the following necessary condition for the existence of positive solution of the system (28) :

[TABLE]

Let $f(\mathbf{x})=\varphi(x_{1}+x_{2})$, where $\varphi$ is a strictly decreasing function defined on the interval $[0,K)$, $0<K\leq\infty$, such that $\varphi(0)=1$ and $\lim_{x\to K}\varphi(x)=0$. Define $m_{1}=\left(\theta+d_{2}-1\right)b_{2}^{-1}$, $m_{2}=\left(\theta^{-1}+d_{2}-1\right)b_{2}^{-1}$, $p_{1}=\varphi^{-1}(m_{1})$, and $p_{2}=\varphi^{-1}(m_{2})$. If (29) holds then the constants $p_{1}$, $p_{2}$ are well defined and $0<p_{1}<p_{2}$. Thus, we find a positive solution of system (28) if and only if (29) holds and

[TABLE]

System (30) has a unique solution

[TABLE]

Since $\alpha>1$, $\theta>1$, and $\beta>0$ we have

[TABLE]

Thus system (28) has a positive solution if and only if (29) holds and

[TABLE]

Now we show how to find parameters $b_{1},b_{2},d_{1},d_{2}$ such that (29) and (32) hold. Assume that $\beta$ is sufficiently small. Since $\gamma=\beta+1-\alpha$, from (18) it follows

[TABLE]

Let $\theta=1+\varepsilon$. Then $\varepsilon=\sqrt{2/(\alpha^{2}-\alpha)}\,\sqrt{\beta}+O(\beta)$ and we can assume that $\varepsilon$ is also sufficiently small. Hence $\theta^{-1}=1-\varepsilon+O(\varepsilon^{2})$, $\alpha\theta+\gamma=1+\alpha\varepsilon+O(\varepsilon^{2})$, $m_{1}=(d_{2}+\varepsilon)b_{2}^{-1}$, and $m_{2}=(d_{2}-\varepsilon)b_{2}^{-1}+O(\varepsilon^{2})$. Assume that $\varphi^{-1}$ is a $C^{2}$-function in a neighbourhood of the point $\bar{x}=d_{2}b_{2}^{-1}$. Then

[TABLE]

where $A=\varphi^{-1}(\bar{x})$ and $B=(\varphi^{-1})^{\prime}(\bar{x})=1/\varphi^{\prime}(A)$. Substituting (33) to (32) we rewrite (32) as

[TABLE]

Taking sufficiently small $\beta$, we are also able to check condition (29). Thus, if $A<-2Bb_{2}^{-1}$, $\beta$ is sufficiently small and $\alpha$ is sufficiently large, both conditions (29) and (32) are fulfilled and a non-trivial periodic solution exists.

Example 1*.*

We consider the two-dimensional model (1) related to the logistic map, i.e., with $f(\mathbf{x})=\varphi(x_{1}+x_{2})$ and $\varphi(x)=1-x/K$ for $x\in[0,K]$ and $\varphi(x)=0$ for $x>K$. Then $\varphi^{-1}(x)=K(1-x)$, and so $A=K(1-d_{2}/b_{2})$, $B=-K$. From (34) it follows that if $2b_{2}/b_{1}<b_{2}-d_{2}<2$, $d_{1}=b_{1}d_{2}/b_{2}-\beta$ and $\beta$ is sufficiently small, then there exists a periodic solution. Let us consider a special example with the following coefficients $b_{1}=2.02$, $b_{2}=0.505$, $d_{1}=0.0399$, $d_{2}=0.01$. Then $b_{1}/d_{1}>b_{2}/d_{2}$, $\alpha=b_{1}/b_{2}=4$, $\beta=\alpha d_{2}-d_{1}=0.0001$, $\gamma=-2.9999$, and $\theta\approx 1.00408$. Then all conditions hold and a positive periodic solution exists. If $K=1$ then the periodic sequence is given by $x_{1}(2n-1)\approx 0.8482$, $x_{1}(2n)\approx 0.8622$, $x_{2}(2n-1)\approx 0.1099$, $x_{2}(2n)\approx 0.1103$ for $n\in\mathbb{N}$. It is interesting that in this case, one-dimensional models (i.e., with the birth and death coefficients $b_{1},d_{1}$ or $b_{2},d_{2}$) have positive and globally stable fixed points because $b_{i}<2+d_{i}$ for $i=1,2$ (see Section 3.4). Hence the two-dimensional model has a locally stable fixed point $(1-d_{1}/b_{1},0)$ but this point is not globally stable.

Example 2*.*

We consider now the two-dimensional Beverton-Holt model with harvesting, i.e., a model of type (1) with $f(\mathbf{x})=\varphi(x_{1}+x_{2})$ and $\varphi(x)=c/(c+x)$ for $x\in[0,\infty)$, $c>0$. A one-dimensional version of this model always has one positive fixed point and this point is globally asymptotically stable (see Section 3.4). We have $\varphi^{-1}(x)=c/x-c$, and so $A=c(b_{2}/d_{2}-1)$, $B=-cb_{2}^{2}/d_{2}^{2}$. Inequality (34) takes the form

[TABLE]

The first inequality is automatically fulfilled for sufficiently small $\beta$. The second inequality holds if $b_{1}>\dfrac{2b_{2}^{2}}{(b_{2}-d_{2})d_{2}}$ and $\beta$ is sufficiently small and then a positive periodic solution exists.

Example 3*.*

We consider now the two-dimensional model (1) related to the Ricker map, i.e., with $f(\mathbf{x})=\varphi(x_{1}+x_{2})$ and $\varphi(x)=e^{-cx}$ for $x\in[0,\infty)$. We have $\varphi^{-1}(x)=-c^{-1}\ln x$, $(\varphi^{-1})^{\prime}(x)=-(cx)^{-1}$ and so $A=c^{-1}\ln(b_{2}/d_{2})$, $B=-b_{2}/(cd_{2})$. Inequality (34) takes the form

[TABLE]

Thus if $d_{2}e^{2/(\alpha d_{2})}<b_{2}<d_{2}e^{2/d_{2}}$ and $\beta$ is sufficiently small, then a positive periodic solution exists. Now we give an example when $T$ have a positive periodic point and both one-dimensional models have globally stable fixed points, i.e., $b_{r}<d_{r}e^{2/d_{r}}$ holds (see Section 3.4). Let $b_{1}=1.0001e^{2}$, $b_{2}=b_{1}/4$, $d_{1}=0.9999$, $d_{2}=0.25$. The coefficients $\alpha=b_{1}/b_{2}=4$, $\beta=\alpha d_{2}-d_{1}=0.0001$, $\gamma=-2.9999$ are the same as in Example  1. Thus $\theta\approx 1.00408$, $\theta^{-1}\approx 0.99594$, and $\alpha\theta+\gamma=1.01642$. For $c=1$ we have $p_{i}=-\ln m_{i}$ for $i=1,2$ and we can check that the periodic sequence is given by $x_{1}(2n-1)\approx 1.49009$, $x_{1}(2n)\approx 1.51455$, $x_{2}(2n-1)\approx 0.000868$, $x_{2}(2n)\approx 0.000871$ for $n\in\mathbb{N}$.

Remark 3*.*

We have restricted examples only to $f$ of the form $f(\mathbf{x})=\varphi(x_{1}+x_{2})$ with the typical $\varphi$ used in the classic competition models, to show that these models can have no coexistence equilibrium, but they can have a positive periodic solution. Formula (28) can be used to find periodic solutions of models with other $f$’s.

3.4. Stability of fixed points

In the previous sections we use some results concerning local and global stability of the transformation $T$ and to make our exposition self-contained we add a section concerning this subject. First we look for fixed points of the transformation $T$ and check their local stability. We assume that $f(\mathbf{0})=1$ and

[TABLE]

Let $\mathbf{x}^{*}$ be a fixed point of $T$, i.e., $T({\mathbf{x}}^{*})={\mathbf{x}}^{*}$. Then $x^{*}_{i}=0$ or $f(\mathbf{x}^{*})=d_{i}/b_{i}=L_{i}^{-1}$ for $i=1,\dots,k$. From (35) it follows that $\mathbf{x}^{*}$ is a fixed point of $T$ if $\mathbf{x}^{*}=\mathbf{x}^{*}_{0}=\mathbf{0}$ or $\mathbf{x}^{*}=\mathbf{x}^{*}_{r}=(0,\dots,x^{*}_{r},\dots,0)$, where $r\in\{1,\dots,k\}$ and $f(\mathbf{x}^{*}_{r})=L_{r}^{-1}$. We assume that the functions $x\mapsto f(0,\dots,x,\dots,0)$ are strictly decreasing. Then $T$ has exactly $k+1$ fixed points $\mathbf{x}^{*}_{r}$, $r=0,\dots,k$. Let $A_{r}$ be the matrix with $a_{ij}^{r}=\dfrac{\partial T_{i}}{\partial x_{j}}({\mathbf{x}_{r}^{*}})$. We have

[TABLE]

where $\delta_{ii}=1$ and $\delta_{ij}=0$ if $i\neq j$. Since $f(\mathbf{0})=1$ we obtain $a_{ii}^{0}=1-d_{i}+b_{i}>1$ and $a_{ij}^{0}=0$ if $i\neq j$, and therefore $\mathbf{x}_{0}^{*}=\mathbf{0}$ is a repulsive fixed point. Now we consider a point $\mathbf{x}_{r}$ with $r>0$. Then $f(\mathbf{x}_{r})=d_{r}/b_{r}$ and from (36) we obtain

[TABLE]

The matrix $A_{r}$ has $k$-eigenvalues $\lambda_{i}$, $i=1,\dots,k$ and $\lambda_{i}=a^{r}_{ii}$. We have

[TABLE]

Observe that if $r=1$ then $\lambda_{i}\in(0,1)$ for $i>1$. If we assume that $-2<b_{1}x_{1}\dfrac{\partial f}{\partial x_{1}}(\mathbf{x}_{1}^{*})<0$, then also $\lambda_{1}\in(0,1)$ and the fixed point $\mathbf{x}_{1}^{*}$ is locally asymptotically stable. If $r>1$ then $\lambda_{i}>1$ for $i<r$ and, consequently, the fixed point $\mathbf{x}_{1}^{*}$ is not asymptotically stable. But if $-2<b_{r}x_{r}\dfrac{\partial f}{\partial x_{r}}(\mathbf{x}_{r}^{*})<0$, the point $\mathbf{x}_{r}^{*}$ is locally semi-stable, i.e., is stable for the transformation $T$ restricted to the set

[TABLE]

In the case of the logistic map $f(\mathbf{x})=1-(x_{1}+\dots+x_{k})/K$ we have $x_{r}=K(1-d_{r}/b_{r})$, $\dfrac{\partial f}{\partial x_{r}}(\mathbf{x}_{r}^{*})=-1/K$, and conditions for stability (or semi-stability) of $\mathbf{x}_{r}^{*}$ reduce to $b_{r}<2+d_{r}$. If the positive fixed point $x^{*}$ of a one-dimensional logistic map is locally asymptotically stable then this point is globally stable, i.e., $T^{n}(x)\to x^{*}$, for $x\in(0,K)$. Thus, Example 1 shows that the behaviour of a $k$-dimensional logistic map and its one-dimensional restrictions can be different. It can have a locally stable fixed point $\mathbf{x}_{1}^{*}$ but this point can be not globally asymptotically stable.

Consider a model with the Beverton-Holt birth rate

[TABLE]

Then we have $x_{r}=c\left(\dfrac{b_{r}}{d_{r}}-1\right)$, $\dfrac{\partial f}{\partial x_{r}}(\mathbf{x}_{r}^{*})=-\dfrac{c}{(c+x_{r})^{2}}$. Conditions for stability (or semi-stability) of $x_{r}$ reduce to the inequality $b_{r}x_{r}c<2(c+x_{r})^{2}$ or, equivalently to $b_{r}^{2}/d_{r}^{2}>b_{r}^{2}/d_{r}-b_{r}$, which always holds because $0\leq d_{r}\leq 1$. The positive fixed point $x^{*}$ of the one-dimensional map $T$ is globally stable because $x<T(x)<x^{*}$ for $x\in(0,x^{*})$ and $T(x)<x$ for $x>x^{*}$. Example 2 shows that a two-dimensional map have a locally stable fixed point $\mathbf{x}_{1}$ but this point can be not globally asymptotically stable.

In the case of the Ricker map $f(\mathbf{x})=e^{-c(x_{1}+\dots+x_{k})}$ we have $x_{r}=\dfrac{1}{c}\ln\dfrac{b_{r}}{d_{r}}$, $\dfrac{\partial f}{\partial x_{r}}(\mathbf{x}_{r}^{*})=-c\dfrac{d_{r}}{b_{r}}$, and conditions for stability (or semi-stability) of $x_{r}$ reduce to $cd_{r}x_{r}>2$, which takes place when $b_{r}<d_{r}e^{2/d_{r}}$. The last inequality is also sufficient for global stability of the fixed point (see e.g. [16, Th. 9.16]).

4. Conclusion

In this paper we consider a discrete time strong competition model. While in its continuous counterpart a population having the maximal turnover coefficient drives the other to extinction, the discrete time model can have no this property. We give sufficient conditions for competitive exclusion for a discrete model. Although this model does not have a coexistence equilibrium, it can have a positive periodic solution. It is interesting that this periodic solution can exist in the case when the restrictions of the model to one dimensional cases have globally stable stationary solutions. Theorem 1 can be also applied to models when the suppression function $f$ depends on other factors, for example the suppression function $f$ can include resource density. It would be interesting to generalize Theorem 1 to models with weaker competition, i.e., when the suppression function is not identical for all subpopulations, or to discrete-continuous hybrid models [8, 10] or to equations on time scales [5].

Acknowledgments

The author is grateful to Dr. Magdalena Nockowska for several helpful discussions while this work was in progress. This research was partially supported by the National Science Centre (Poland) Grant No. 2014/13/B/ST1/00224.