On permanence of all subsystems of competitive Lotka–Volterra systems with delays

Abstract

In this paper, permanence for a class of competitive Lotka–Volterra systems is considered that have distributed delays and constant coefficients on interaction terms and have time dependent growth rate vectors with an asymptotic average. A computable necessary and sufficient condition is found for the permanence of all subsystems of the system and its small perturbations on the interaction matrix. This is a generalization from systems without delays to delayed systems of Ahmad and Lazer’s work on total permanence (S. Ahmad, A.C. Lazer, Average growth and total permanence in a competitive Lotka–Volterra system, Ann. Mat. 185 (2006) S47–S67). In addition to Ahmad and Lazer’s example showing that permanence does not imply total permanence, another example of permanent system is given having a non-permanent subsystem. As a particular case, a necessary and sufficient condition is given for all subsystems of the corresponding autonomous system to be permanent. As this condition does not rely on the delays, it actually shows the equivalence between such permanence of systems with delays and that of corresponding systems without delays. Moreover, this permanence property is still retained by systems as a small perturbation of the original system.

Introduction

In the context of differential equations, a class of systems of Lotka–Volterra type differential equations have the form

$$x_i^{\prime }\left(t\right)=x_i\left(t\right)f_i(t,x)\text{,}i\in I_N\text{,}$$

where, for any positive integer $m$, $I_m=\{1,2,\dots ,m\}$ and for a fixed positive integer $N$ and each $i\in I_N$, $f_i$ is vaguely described as a continuous function of $(t,x)$ (without delays) or functional (with finite or infinite delays). Since a particular case of (1) was established and analyzed in the 1920s as a mathematical model of population growth of a community of $N$ species, where $x_i\left(t\right)$ denotes the population size of the $i$th species at time $t$, the study of such systems is mainly restricted to the nonnegative cone

$${\mathbb{R}}_{+}^N=\{x\in {\mathbb{R}}^N:\forall i\in I_N,x_i\ge 0\}\text{.}$$

In particular, solutions of (1) are mostly considered in the interior of ${\mathbb{R}}_{+}^N$,

$$\mathrm{int}{\mathbb{R}}_{+}^N=\{x\in {\mathbb{R}}_{+}^N:\forall i\in I_N,x_i>0\}\text{.}$$

Ecologically, one concern about the community is whether all species can coexist uniformly, i.e. each species has an ultimate positive lower bound and an upper bound regardless of its starting population. This is mathematically gauged by the concept of permanence. A system is called permanent if there are $r>0$ and $M>r$ such that every solution of the system in $\mathrm{int}{\mathbb{R}}_{+}^N$ satisfies $r\le x_i\left(t\right)\le M$ for all $i\in I_N$ and all sufficiently large $t$.

Ahmad and Lazer [1] (or [2]) showed by an example that a permanent system may have non-permanent subsystems (see Section 2 for another example). However, in some situation, a condition for permanence of a system also ensures permanence of all subsystems. Earlier observations on this point can be found in [3]. The main focus of this paper is on a condition for all subsystems to be permanent.

The simplest instance of (1) is the autonomous Lotka–Volterra system of ordinary differential equations having the form

$$x_i^{\prime }=x_i(b_i-A_{i}x)\text{,}i\in I_N\text{,}$$

where $A_i$ is the $i$th row of a matrix $A=\left(a_{ij}\right)$ and the $b_i$ and $a_{ij}$ are constant real numbers. For (2) to be permanent, sufficient conditions as well as necessary conditions can be found in [4], [5]. For autonomous systems of the form $x_i^{\prime }=x_i{f}_i\left(x\right)$, Schreiber [6] investigated $C^r$ robust permanence (i.e., the permanence of the system and its sufficiently small $C^r$ perturbations). Based on this theory, Mierczyński and Schreiber [7] obtained necessary and sufficient conditions for all subsystems of a Kolmogorov system (including the system itself) to be $\left(C^1\right)$ robustly permanent. The following theorem is the result of a simple application of one of the conditions to (2).

Theorem 1 [7, Corollary 3.1]

Assume (2) is dissipative. Then all subsystems of (2) are robustly permanent if and only if for every $I\subset I_N$there is a unique equilibrium $x_I^{\ast }$ in

$$C_I^0=\{x\in {\mathbb{R}}_{+}^N:x_j>0\text{<mi mathvariant="italic" is="true">if and only if</mi>}j\in I_N\setminus I\}$$

such that $A_i{x}_I^{\ast }<b_i$ for all $i\in I$.

Parallel to system (2), the nonautonomous instances of (1) without delays are

$$x_i^{\prime }\left(t\right)=x_i\left(t\right)[r_i\left(t\right)-A_{i}x\left(t\right)]\text{,}i\in I_N$$

and

$$x_i^{\prime }\left(t\right)=x_i\left(t\right)[{\tilde{r}}_i\left(t\right)-A_i\left(t\right)x\left(t\right)]\text{,}i\in I_N\text{,}$$

where the $r_i\left(t\right)$ and ${\tilde{r}}_i\left(t\right)$ are continuous functions from ${\mathbb{R}}_0=(-\infty ,\infty )$, $(c,\infty )$ or $[c,\infty )$ for some $c\in \mathbb{R}$ to $\mathbb{R}$, the $A_i$ are the same as in (2) and the $A_i\left(t\right)=(a_{i1}\left(t\right),\dots ,a_{iN}\left(t\right))$ are continuous from ${\mathbb{R}}_0$ to ${\mathbb{R}}_{+}^N$. Although there are many investigations dealing with permanence related problems of (4), (5) and their variations, we only quote [8], [9], [1], [10], [11], [12], [13], [14], [15] and the references therein as examples, with most of them providing sufficient or necessary conditions for permanence. A commonly used technique is the employment of the average

$$m(g,t_1,t_2)=\frac{1}{t_2-t_1}{\int }_{t_1}^{t_2}g\left(t\right)dt$$

of a function $g$. Ahmad and Lazer [11] investigated the permanence of competitive systems (4), (5) under the assumptions that the $r_i\left(t\right)$, ${\tilde{r}}_i\left(t\right)$ and $a_{ii}\left(t\right)$ have positive lower and upper bounds, that the $a_{ij}\left(t\right)$ are bounded and nonnegative, and that $r:{\mathbb{R}}_0\to {\mathbb{R}}_{+}^N$ satisfies

$$\forall i\in I_N\text{,}\underset{t\to \infty }{lim}m(r_i,t_0,t)=b_i\text{uniformly for}{t}_0\in {\mathbb{R}}_0\text{.}$$

It is noted from [11] that a large class of functions, including periodic and almost periodic functions, satisfy (7). System (5) can be viewed as an $\epsilon$-perturbation of (4) if

$$\forall i,j\in I_N,\forall t\in {\mathbb{R}}_0\text{,}|{\tilde{r}}_i\left(t\right)-r_i\left(t\right)|\le \epsilon \text{,}|a_{ij}\left(t\right)-a_{ij}|\le \epsilon \text{.}$$

Ahmad and Lazer [11] introduced the concept of total permanence of (4) as the existence of $\epsilon >0$ such that all subsystems of (5) satisfying (8) are permanent. Obviously, the example given in [1] or [2] shows that permanence does not imply total permanence. As an analogue of Theorem 1 for (2), we have the following nice result for (4), (5).

Theorem 2 [11]

System (4) with (7), $b={(b_1,\dots ,b_N)}^T\in \mathrm{int}{\mathbb{R}}_{+}^N$, $A\in {\mathbb{R}}_{+}^{N\times N}$, $a_{ii}>0$ for $i\in I_N$ is totally permanent if and only if $b$ and $A$ satisfy the $(I-J)$-conditions (to be defined in Section  3).

The aim of this paper is to investigate permanence of competitive Lotka–Volterra systems with delays and extend Theorem 1, Theorem 2 to such systems.

As another instance of (1) we consider the following autonomous system with distributed delays,

$$x_i^{\prime }\left(t\right)=x_i\left(t\right)[b_i-\sum _{j=1}^N{a}_{ij}{\int }_{-\tau }^0{x}_j(t+\theta )d{\xi }_{ij}\left(\theta \right)]\text{,}i\in I_N\text{,}$$

where the $b_i$ and $a_{ij}$ are real valued constants, $\tau >0$, and the ${\xi }_{ij}$ are nondecreasing functions from any open interval containing $[-\tau ,0]$ to ${\mathbb{R}}_{+}$ satisfying ${\xi }_{ij}\left(0^{+}\right)-{\xi }_{ij}(-{\tau }^{-})=1$ for all $i,j\in I_N$. For some particular cases of (9) when $N=2$, some necessary, sufficient or necessary and sufficient conditions for the permanence of (9) can be found in [16], [17], [18], [19], [20], [21], [22] and the references therein. Kuang observed the relationship between (2), (9) and made the conjecture that system (9) is permanent if and only if (2) is permanent. Chen, Lu and Wang [21] showed by examples that this conjecture is not true for cooperative systems. Is it true for competitive systems and predator–prey systems? We cannot answer this question directly but, indirectly, we shall see that the property of (2) secured by Theorem 1 is not affected by the delays in (9).

Finally, another large class of systems as instances of (1) are nonautonomous systems with delays. Consider the systems

$$x_i^{\prime }\left(t\right)=x_i\left(t\right)[r_i\left(t\right)-\sum _{j=1}^N{a}_{ij}{\int }_{-\tau }^0{x}_j(t+\theta )d{\xi }_{ij}\left(\theta \right)]\text{,}i\in I_N$$

and

$$x_i^{\prime }\left(t\right)=x_i\left(t\right)[{\tilde{r}}_i\left(t\right)-\sum _{j=1}^N{\int }_{-\tau }^0{x}_j(t+\theta )d_{\theta }{\eta }_{ij}(t,\theta )]\text{,}i\in I_N\text{,}$$

where the $r_i\left(t\right)$ and ${\tilde{r}}_i\left(t\right)$ are the same as in (4), (5), the $a_{ij}$ and ${\xi }_{ij}$ are the same as in (9), and each ${\eta }_{ij}(t,\theta )$ is of bounded variation in $\theta \in [-\tau ,0]$ for each fixed $t\in {\mathbb{R}}_0$ and ${\int }_{-\tau }^0\phi \left(\theta \right)d_{\theta }{\eta }_{ij}(t,\theta )$ is continuous in $t$ for each fixed $\phi \in C([-\tau ,0],\mathbb{R})$. Various conditions for permanence of (10), (11) or systems of other forms with delays can be found in [23], [24], [25], [22], [26], [27], [28], [29] and the related references quoted there. In this paper, we shall extend Theorem 2 for (4) to systems (10), (11). This will effectively show that the property of (4) described by Theorem 2 is independent of the delays in (10), (11).