Global stability and persistence of complex foodwebs

Abstract

We develop a novel approach to study the global behaviour of large foodwebs for ecosystems where several species share multiple resources. The model extends and generalizes some previous works and takes into account self-limitation. Under certain explicit conditions, we establish the global convergence and persistence of solutions.

1 Introduction

To mathematically show the existence and stability of large foodwebs, large and complex foodwebs in nature are still one of the key problems in theoretical ecology. A specific part of this theoretical issue is that many species can share more than just a few resources (for example ocean ecosystems including thousands of phytoplankton species) yet the competitive exclusion principle [8, 24] asserts that such foodwebs should not exist. To partly explain that paradox [13] showed that a system consisting of a single resource and three species can support chaotic dynamics where all species coexist, another explanation of the paradox was proposed in [21] where self-limitation effects have been taken into account.

In this paper, we add complexity to the work of [13, 21] by extending the dynamical equations considered in [21] with self-limitation effects [1,2,3, 16, 17, 21]) (see also a turbidostat model in [18]). We obtain a complete description of the large time behaviour of the system. In particular, we explore the range in the parameter space that leads the system to the global stable equilibrium point, see the explicit estimates in (37) and (38). Furthermore, we show that if the self-limitations exceed some critical values then the system exhibits either global stability or persistence, see Propositions 9 and  10.

Traditionally, the Lyapunov function approach is used to establish global stability, see a recent review in [10]. In our case, however, an explicit information about equilibrium points is not available. Instead, we transform our problem to a finite-dimensional nonlinear fixed point problem for an appropriate non-increasing operator. We show that the asymptotic behaviour of a generic solution to the initial problem is well controlled by iterations of the introduced operator. This allows us to derive explicit a priori estimates (see Theorem 1) and the global stability.

The paper is organized as follows. In Sect. 2, we present the model with self-limitations, and in Sect. 3, we obtain some preliminary results. We review some elementary facts on period-two-points of non-increasing maps in Sect. 4 and discuss the structure and stratification of equilibrium points in Sect. 5. In particular, in Sect. 6 we consider the so-called special equilibrium points which are significant for the large time behaviour of the original dynamical system. Here we also define the corresponding finite-dimensional fixed point problem. To study its dynamics and convergence, we need to suitably polarize the fixed point problem. This allows us to establish bilateral estimates for the corresponding $\omega$-limit set. The main result of this section is contained in Proposition 7, which gives a sufficient condition for the existence of a unique fixed point. In Sect. 7, we return to the main dynamical system formulate and prove the main results on the large behaviour of the original dynamic system. In particular, we obtain some explicit conditions when the system obeys strong persistence. In Sect. 8, we briefly discuss our results and relate them to some previous research. We finally mention that in our recent paper [14], we apply the results of the present paper to obtain explicit estimates of biodiversity for competition systems with extinctions.

2 The model

Given $x,y\in {\mathbb {R}}^n$, we use the standard vector order relation: $x\le y$ if $x_i\le y_i$ for all $1\le i\le n$, $x< y$ if $x\le y$ and $x\ne y$, and $x\ll y$ if $x_i< y_i$ for all i; ${\mathbb R}_{+}^n$ denotes the nonnegative cone $\{x\in {\mathbb R}^n:x\ge 0\}$ and for $a\le b$, $a,b\in {\mathbb R}^n$

$$\begin{aligned}{}[a,b]=\{x\in {\mathbb R}^{n}:a\le x\le b\} \end{aligned}$$

is the closed box with vertices at a and b.

We consider the model where the population dynamics of M species competing for m complementary resources is governed by chemostat-like equations

$$\begin{aligned}&\displaystyle \frac{\mathrm{d}x_j}{\mathrm{d}t}=x_j ( \phi _j(v)- \mu _j - \gamma _{j} \; x_j), \quad j=1,\ldots , M \end{aligned}$$

(1)

$$\begin{aligned}&\displaystyle \frac{\mathrm{d}v_i}{\mathrm{d}t}=D_i(S_i -v_i) - \sum _{j=1}^M c_{ ij} \; x_j \; \phi _j(v), \quad i=1,\ldots , m \end{aligned}$$

(2)

$$\begin{aligned}&\displaystyle x(0)\gg 0, \qquad 0\le v(0)\le S:=(S_1, \ldots , S_m), \end{aligned}$$

(3)

Here $x_j(t)$ is species j abundance and $v_i(t)$ is the concentration of resource i at time t, $\mu _j$ are the species mortalities, $S_i$ is the supply of resource i, $c_{ij} >0$ is the content of resource i in species j (growth yield constants), $D_i$ is the rate of exchange of resource i, resource turnover (or dilution) rate), $\gamma _{j} >0$ is a self-limitation constant of species j. We shall assume that the specific growth rates $\phi _j$ are bounded Lipschitz functions subject to the following standard conditions:

$$\begin{aligned}&\displaystyle \phi _j(v) =0 \,\,\Leftrightarrow \,\, v \in \partial {\mathbb R}^m_{+}; \end{aligned}$$

(4)

$$\begin{aligned}&\displaystyle \phi _j(v) \text {is a non-decreasing function of each} v_i. \end{aligned}$$

(5)

The most relevant for biological applications example of the specific growth functions $\phi _j$ is given by the Monod equation and Liebig’s ‘law of the minimum’

$$\begin{aligned} \phi _j(v) =\min \left( \frac{r_jv_1}{K_{1j} + v_1}, \ldots , \frac{r_jv_m}{K_{mj} + v_m} \right) , \end{aligned}$$

(6)

where $r_j$ is the maximum specific growth rate of species j and $K_{ij}$ is the half-saturation constant for resource i of species j. Obviously, the functions (6) meet the above conditions.

In the absence of self-limitation ($\gamma _j=0$), the present model naturally appears in the bioengineering context [4] and was extensively studied for $m\le 2$ resources for both equal resource turnover rates $\mu _j=D_i=D$ in [11,12,13] and for different removal rates $\mu _j$ in [9, 10, 20, 23], see also a recent review in [22]. For a single resource $m=1$, the dynamics of the standard model in the absence of self-limitation is completely determined by the break-even concentrations $R_j$ defined as $\phi _j(R_j)=\mu _j$, see [4, 10]. For example, if the lowest break-even concentration

$$\begin{aligned} R^*=\min \{\,R_1\,\ldots , R_m\} \end{aligned}$$

(7)

achieves on a single species k then

$$\begin{aligned} \lim \limits _{t\rightarrow \infty } v_1(t)=R^*, \quad \lim \limits _{t\rightarrow \infty } x_{k}(t)=\frac{1}{\mu _1}(S_1-R^*) \end{aligned}$$

while $\lim _{t\rightarrow \infty } x_j(t)=0$ for all $j\ne k$. However, if $m\ge 3$, the behaviour becomes much more involved. Recent numerical simulations [12, 13] strongly support the possible chaos scenario for $(m,M)=(3,6)$ or (5, 6). An important step was done by Li [19] who established the existence of the limit cycle for $m=M=3$.

3 Preliminaries

In what follows, we shall assume that $\gamma _j>0$.

Proposition 1

Solution (x(t), v(t)) of (1), (2), (3) is well-defined and bounded for all $t\ge 0$ and

$$\begin{aligned}&0\le x_j(t) \le x_j(0) \left( e^{(\mu _j-\phi _j(S)) t} +\frac{1-e^{(\mu _j-\phi _j(S)) t}}{\phi _j(S) - \mu _j}\gamma _j x_j(0)\right) ^{-1}, \end{aligned}$$

(8)

$$\begin{aligned}&0\le v_i(t) \le S_i (1- e^{-D_it}) + v_i(0) e^{-D_it}. \end{aligned}$$

(9)

If $\phi _j(S)=\mu _j$, (8) should be replaced by $0\le x_j(t) \le \frac{x_j(0)}{1+ \gamma _j x_j(0)t}$. In particular, [0, S] is an invariant subset. Furthermore, if $\phi _j(S)\le \mu _j$ for some i then $\lim _{t\rightarrow \infty }x_j(t)=0.$

Proof

Note that by (1), $x_j(t)$ never vanishes unless $x_j(t)\equiv 0$. In particular, by (1) $x(t)\gg 0$ as long as x(t) is defined. Furthermore, if $h_i(x,v)$ denote the right-hand side of (2) then by (4) $h_i(x,v)=D_iS_i>0$ for any $v\in \partial {\mathbb R}^m_{+}$, thus $v_i(0)\ge 0$ implies that $v_i(t)>0$ for all admissible $t>0$, see Proposition  2.1 in [6]. Similarly, $h_i(x,S)< 0$ (unless $x=0$) and $v(0)\le S$ yields $v(t)\le S$, and thus (3) and (5) imply $0\le \phi _j(v) \le \phi _j(S)$. This proves that ${\mathbb R}^{M}_+\times [0,S]$ is an invariant subset for (1), (2), (3). Furthermore, $x_j(t) \le y_i(t)$, where $y_i(t)$ is the solution of the Cauchy problem

$$\begin{aligned} \frac{\mathrm{d}y_i}{\mathrm{d}t}=y_i( \phi _j(S) -\mu _j- \gamma _j y_i), \quad y_i(0)=x_j(0), \quad 1\le i\le M. \end{aligned}$$

This readily yields (8) and $\sum _{i=1}^M c_{ki} x_j\phi _j(v)\ge 0$ yields the upper estimate in (9). Since (x(t), v(t)) is a bounded solution, it is well-defined for all $t\ge 0$. Finally, if $\phi _j(S)\le \mu _j$ then (8) implies $\lim _{t\rightarrow \infty }x_j(t)=0$. $\square$

Proposition 1 shows that the extinction dynamics of (1), (2), (3) depends on the sign of $\phi _j(S)-\mu _j$: for species i to survive, its specific growth rate $\phi _j(S)$ at the supply point S must exceed its specific mortality rate $\mu _j$. To eliminate the trivial extinctions, we shall assume in what follows that the survivability condition holds:

$$\begin{aligned} \quad \phi _j(S)>\mu _j\quad \text { for all }j. \end{aligned}$$

(10)

For the Monod–Liebig model (6), the survivability condition (10) is equivalent to $0\ll R^{(j)}\ll S,$ where $R^{(j)}:=(R_{1j},\ldots , R_{mj})$ and $R_{ij}:=\frac{\mu _j}{r_j-\mu _j}K_{ij}$ are the resource requirement of a species j for a resource i [12].

Below we summarize some elementary observations which will be used throughout the paper.

Lemma 1

Let $f(x),g(x)\not \equiv 0$ be continuous nonnegative and non-decreasing maps $[0,S]\rightarrow {\mathbb R}{}$, $f(0)=0$, where $S>0$ is a real number. Then $S-x=f(x)$ has a unique solution $0<x_f<S$. If $f(x)\ge g(x)$ ($f(x)> g(x)$ resp.) then $x_f\le x_g$ ($x_f< x_g$ resp).

Proof

An idea of the proof is clear from the figure below.

$\square$

Lemma 2

Let $v'(t)=F(v(t),t)$ and ${\tilde{v}}'(t)={\widetilde{F}}({\tilde{v}}(t),t)$, $t\in [0,T]$, where F(z, t) and ${\tilde{F}}(z,t)$ are decreasing functions of z for each t, $F(z,t)\ge {\widetilde{F}}(z,t)$ and $v(0)\ge {\tilde{v}}(0)$. Then $v(t)\ge {\tilde{v}}(t)$ for all $t\in [0,T]$.

Proof

Let $u(t)={\tilde{v}}(t)-v(t)$, then $u(0)=0$. If there exists $\xi >0$ such that $u(\xi )>0$ then

$$\begin{aligned} u'(\xi )={\widetilde{F}}({\tilde{v}}(\xi ),\xi )-F(v(\xi ),\xi )< {\widetilde{F}}(v(\xi ),\xi )-F(v(\xi ),\xi )\le 0. \end{aligned}$$

Since $u(0)\le 0$, $u(\xi )>0$ and $u'(\xi )<0$, u(t) has a local maximum in $(0,\xi )$. Let $0<\eta <\xi$ be a maximum point. Then $u(\eta )>0$ and $u'(\eta )=0$, i.e. ${\tilde{v}}(\eta )>v(\eta )$ and

$$\begin{aligned} {\widetilde{F}}({\tilde{v}}(\eta ),\eta )={\tilde{v}}'(\eta ) =v'(\eta )=F(v(\eta ),\eta )> F({\tilde{v}}(\eta ),\eta )\ge {\widetilde{F}}({\tilde{v}}(\eta ),\eta ), \end{aligned}$$

a contradiction follows. $\square$

Lemma 3

Let F(z, t) be Lipschitz function in $[0,S]\times [0,\infty )$ such that

• (a) $F(0,t)<0$, $F(S,t)>0$ for all $t>0$;

• (b) there exists $c>0$ such that

  $$\begin{aligned} F(z_1,t)-F(z_2,t)\ge c(z_2-z_1) \quad \text {for}\;t\ge 0\;\text {and}\;0\le z_1<z_2\le S; \end{aligned}$$

• (c) if $0<z(t)<S$ is the unique solution of $F(z(t),t)=0$ then $\lim \limits _{t\rightarrow \infty }z(t)={\bar{z}}$.

Then for any solution of

$$\begin{aligned} u'(t)=F(u(t),t), \quad 0<u(0)<S \end{aligned}$$

there holds $\lim \limits _{t\rightarrow \infty }u(t)={\bar{z}}$.

Proof

By (b) F(z, t) is strictly decreasing in z for each $t\ge 0$. It follows from the conditions (a)–(b) and the classical Clarke result [5] that z(t) in (c) is well-defined and local Lipschitz on $[0,\infty )$. It follows from (a) that $0<u(t)<S$ for all $t\ge 0$. Now, two alternatives are possible: (i) either there exists $T>0$ such that $u(t)\ne z(t)$ for $t\ge T$, or (ii) there exists $t_k\nearrow \infty$: $u(t_k)=z(t_k)$. First let (i) hold and assume without loss of generality that $u(t)<z(t)$ for $t\ge T$. Then

$$\begin{aligned} u'(t)=F(u(t),t)-F(z(t),t)\ge c(z(t)-u(t))\ge 0, \end{aligned}$$

(11)

hence u(t) is non-decreasing, therefore there exists

$$\begin{aligned} {\bar{u}}:=\lim _{t\rightarrow \infty } u(t)\le \lim _{t\rightarrow \infty } z(t)={\bar{z}}. \end{aligned}$$

(12)

Combining (12) with the monotonicity of u(t) and (11) implies

$$\begin{aligned} \int _{t}^\infty |z(s)-u(s)|\,\mathrm{d}s\le \frac{1}{c}({\bar{u}}-u(t))\rightarrow 0\,\, \text { as }t\rightarrow \infty \end{aligned}$$

which implies the equality in (12). Next, if (ii) holds then $\lim _{k\rightarrow \infty }u(t_k)={\bar{z}}$. Assume by contradiction that, for example, ${\bar{u}}:=\lim \sup _{t\rightarrow \infty } u(t)>{\bar{z}}$ and let $\xi _k\nearrow \infty$ be a corresponding sequence where the $\lim \sup$ is attained. Since

$$\begin{aligned} \lim _{k\rightarrow \infty }u(t_k)={\bar{z}}<{\bar{u}}=\lim _{k\rightarrow \infty }u(\xi _k) \end{aligned}$$

one can redefine the sequence $\xi _k$ such that each $\xi _k$ becomes a local maximum of u. This yields $0=u'(\xi _k)=F(u(\xi _k),\xi _k)$, thus $u(\xi _k)=z(\xi _k)$. Passing to limit as $k\rightarrow \infty$ yields a contradiction. $\square$

4 Period-two-points of non-increasing maps

Let $0\in D\subset {\mathbb R}^{n}_+$ and $G:D\rightarrow D$ be an arbitrary map. Recall that a pair (a, b), $a,b\in D$, is called a period-two-point [7, p. 387], or $(a,b)\in {{\,\mathrm{Fix}\,}}_2(G)$, if

$$\begin{aligned} G(a)=b, \quad G(b)=a. \end{aligned}$$

(13)

Any fixed point $c\in {{\,\mathrm{Fix}\,}}(G)$ gives rise to a trivial period-two-point (c, c).

Hereinafter, we assume that G is continuous and non-increasing in D, i.e. $G(x)\ge G(y)$ for any $x\le y$ in D. Note that G is then automatically bounded:

$$\begin{aligned} 0\le G(x)\le G(0),\qquad \forall x\in D. \end{aligned}$$

(14)

Since $0\in D$, the iterations $u^0=0$, $u^k:=G^k(0)\in D$, $k\ge 1$, are well-defined, $u^1\ge u^0=0$ (an a priori estimate) and $u^2=G(u^1)\le G(u^0)=u^1$ (by virtue of the monotonicity of G). Hence, it follows by induction that

$$\begin{aligned} u^0\le u^2\le \cdots u^{2k}\le \cdots u^{2k+1}\le \cdots \le u^3\le u^1. \end{aligned}$$

(15)

This implies that the limits

$$\begin{aligned} \check{0}_G:=\lim _{k\rightarrow \infty }u^{2k}\le {\hat{0}}_G:=\lim _{k\rightarrow \infty }u^{2k-1} \end{aligned}$$

(16)

exist and $(\check{0}_G,{\hat{0}}_G)$ is a period-two-point of G:

$$\begin{aligned} G(\check{0}_G)={\hat{0}}_G, \quad \check{0}_G=G({\hat{0}}_G). \end{aligned}$$

(17)

Thus obtained period-two-point is extremal as the following property shows.

Proposition 2

For any $(a,b)\in {{\,\mathrm{Fix}\,}}_2(G)$ there holds

$$\begin{aligned} \check{0}_G\le a,\quad b\le {\hat{0}}_G. \end{aligned}$$

(18)

In particular,

$$\begin{aligned} \check{0}_G\le c\le {\hat{0}}_G, \qquad \forall c\in {{\,\mathrm{Fix}\,}}(G), \end{aligned}$$

(19)

and the box

$$\begin{aligned}{}[\check{0}_G,{\hat{0}}_G]:=\{u:\check{0}_G\le u\le {\hat{0}}_G\} \end{aligned}$$

(20)

is invariant under the mapping G.

Proof

Since $a\ge u^0=0$ and G is a non-increasing, one has

$$\begin{aligned} u^{2k}\le G^{2k}(a)=a, \quad u^{2k-1}\ge G^{2k-1}(a)=b, \quad \text { for all }\,k=1,2,\ldots \end{aligned}$$

This readily yields (18). Then (19) follows from the fact that (c, c) is a period-two-point for any $c\in {{\,\mathrm{Fix}\,}}(G)$. The last claim of the proposition follows immediately from the monotonicity of G and (17). $\square$

Proposition 3

Let $x,y\in D$ be such that

$$\begin{aligned} G(y)\le x,\quad y\le G(x). \end{aligned}$$

(21)

Then there exists $(a,b)\in {{\,\mathrm{Fix}\,}}_2(G)$ such that

$$\begin{aligned} \begin{aligned} a&:=\lim _{k\rightarrow \infty }y^{2k-1}=\lim _{k\rightarrow \infty }x^{2k}\ge \check{0}_G,\\ b&:=\lim _{k\rightarrow \infty }y^{2k}=\lim _{k\rightarrow \infty }x^{2k-1}\le {\hat{0}}_G. \end{aligned} \end{aligned}$$

(22)

Proof

Let $y^0=y$ and $y^k=G^k(y)$, $k\ge 1$, hence (21) becomes

$$\begin{aligned} y^1\le x^0,\quad y^0\le x^1. \end{aligned}$$

Applying G we yields $y^2\ge x^1\ge y^0$ and $x^0\ge y^1\ge x^2$. Proceeding by induction on k, we obtain by virtue of (14)

$$\begin{aligned} 0&\quad \le \cdots \le x^{4}\le y^3\le x^2\le y^1 \le x^0,\\ y^0&\quad \le x^1\le y^2\le x^3\le y^4\le \cdots \le G(0). \end{aligned}$$

This implies the existence of limits in (22). It also follows that $G(a)=b$ and $G(b)=a$, hence $(a,b)\in {{\,\mathrm{Fix}\,}}_2(G)$ and $a\le x$, $y\le b$. Combining with the extremal property (18) yields (22). $\square$

5 Stratification of equilibrium points

Let us denote by E the set of nonnegative equilibrium points (stationary solutions) of (1)–(2). It is natural to consider the standard stratification

$$\begin{aligned} E=\bigcup _{J}E_J, \end{aligned}$$

where

$$\begin{aligned} E_J=\{(x,v)\in E: \,\, x_j\ne 0 \,\,\Leftrightarrow \,\, j\in J \}, \end{aligned}$$

and J runs over all subsets of $\{1,2,\ldots ,M\}$. The supply point S is the equilibrium resource availabilities in the absence of any species and obviously (0, S) is the only point in $E_{\emptyset }$:

$$\begin{aligned} E_{\emptyset }=\{(0,S)\}. \end{aligned}$$

Proposition 4

For an arbitrary $(0,S)\ne (x,v)\in E$ there holds

$$\begin{aligned} x>0 \quad \text {and}\quad 0\ll v\ll S. \end{aligned}$$

(23)

Proof

If $x=0$ then $v=S$, thus $x>0$. If some $v_i=0$ then (4) yields $\phi _j(v)=0$ for all j, hence by (2) $v_i=S_i$, a contradiction, i.e. $v\gg 0$. Finally, note that $v\le S$. If $v_i=S_i$ for some i then $\sum _{j=1}^M c_{ ij} \; x_j \; \phi _j(v)=0$. By the above, there exists $x_{k}\ne 0$, therefore $\phi _k(v)=0$ implying by (4) that $v\in \partial {\mathbb R}^{m}_+$, thus $\phi _j(v)=0$ for all j. Applying the stationary condition to (2) we see that $v=S$, a contradiction with $v\in \partial {\mathbb R}^{m}_+$. Therefore, $v\ll S$. $\square$

Let $(x,v)\in E_J$. Then $x_j=0$ if $j\not \in J$ and

$$\begin{aligned} x_j={\mathbf {X}}_j(v):=\frac{1}{{\gamma _j}} \left( \phi _j(v) -\mu _j\right) _+>0\, \quad \text {for all}\,i\in J, \end{aligned}$$

where $w_+=\max (0,w)$, therefore v is determined uniquely by

$$\begin{aligned} \begin{aligned} v_i&=S_i-\sum _{j\in J} \frac{c_{ij}}{D_i}\, {\mathbf {X}}_j(v) \phi _j(v)=:({\mathbf {F}}_J(v))_i. \end{aligned} \end{aligned}$$

(24)

Extend $\mathbf{F }_J$ by ${\mathbf {F}}_{\emptyset }(v):= S$. In the present setting, if $(x,v)\in E_J$ then v solves the fixed point problem

$$\begin{aligned} v={\mathbf {F}}_J(v), \end{aligned}$$

(25)

and

$$\begin{aligned} x_j= \left\{ \begin{array}{ll} 0&{}\quad \text {if}\;j\not \in J\\ {\mathbf {X}}_j(v)&{}\quad \text {if}\;j\in J\\ \end{array} \right. \end{aligned}$$

(26)

The converse is not necessarily true: if v is a solution of (25) and x is defined by (26) then (x, v) is an equilibrium point in $E_{J'}$ for some $J'\subset J$. Indeed, it might happen that $\phi _j(v)\le \mu _j$, i.e. $x_j=0$ for some $j\in J$. On the other hand, if $J\ne \emptyset$ then necessarily $J'\ne \emptyset$ because if $x_j=0$ for all j then $(x,v)=(0,S)$, but ${\mathbf {F}}_J(S)\ll S$ in view of (4), a contradiction with (25).

To distinguish this situation, we denote by

$$\begin{aligned} {\widetilde{E}}_{J}= \text { the set of solutions }(x,v)\text { of } (25)\text { and }(26). \end{aligned}$$

Then ${\widetilde{E}}_{\emptyset }=E_{\emptyset }$, and the above argument yields that for any $J\ne \emptyset$

$$\begin{aligned} E_{J}\subset {\widetilde{E}}_{J}\subset \bigcup _{\emptyset \ne J'\subset J} E_{J'} \end{aligned}$$

(27)

Thus refined stratification $J\rightarrow {\widetilde{E}}_{J}$ still contains information about all equilibrium points but it has better properties than $J\rightarrow {E}_{J}$.

Proposition 5

For any $J\ne \emptyset$, the set ${\widetilde{E}}_{J}$ is nonempty.

Proof

Consider a modified fixed point problem

$$\begin{aligned} v=({\mathbf {F}}_J(v))_+:=\max ({\mathbf {F}}_J(v),0). \end{aligned}$$

Then $v\rightarrow ({\mathbf {F}}_J(v))_+$ maps continuously the box [0, S] into itself, hence by Brouwer’s theorem there exists a fixed point $v\in [0,S]$. If $v_k=0$ for some k then by (4) we have $\phi _j(v)=0$ for all j, thus $v_k=({\mathbf {F}}_J(v))_k=S_k$, a contradiction. Thus $v\gg 0$ and $v_k=[{\mathbf {F}}_k(v)]_+>0$ for all k, therefore in fact $v_k={\mathbf {F}}_k(v)$ holds for all k. This proves that v is a solution of the original fixed point problem (25) and $v\gg 0$. If x is defined by (26) then it follows that $(x,v)\in {\widetilde{E}}_J$. $\square$

6 An auxiliary finite-dimensional fixed point problem

Among all equilibrium points in E, we shall distinguish the special ones, namely those contained in

$$\begin{aligned} {\widetilde{E}}_{M}:={\widetilde{E}}_{\{1,2,\ldots , M\}}. \end{aligned}$$

Equivalently, a point (x, v) is said to be a special (equilibrium) point if and only if v is a solution of the fixed point problem

$$\begin{aligned} v={\mathbf {F}}(v),\quad {\mathbf {F}}:={\mathbf {F}}_{\{1,2,\ldots , M\}}, \end{aligned}$$

(28)

and x is given by

$$\begin{aligned} x_j={\mathbf {X}}_j(v):=\frac{1}{{\gamma _j}} (\phi _j(v) -\mu _j)_{+}. \end{aligned}$$

(29)

By Proposition 5, the set of special equilibrium points is nonempty. Note also that if (x, v) is an arbitrary equilibrium point of (1)–(2) with $x\gg 0$ then it is necessarily a special one because by (1) $\phi _j(v)>\mu _j$ for all j, hence x is determined by (29) and therefore v satisfies (28).

The set of special equilibrium points ${\widetilde{E}}_{M}={{\,\mathrm{Fix}\,}}({\mathbf {F}})$ reflects the complexity of large-time dynamics of the original system in the following sense. Theorem 1 shows that if there exists a unique global stable equilibrium point of (1), (2), (3) then it is necessarily a special point (in this case, obviously, unique). Therefore, the structure and the number of special equilibrium points play a crucial role in the large-time dynamics of (1), (2), (3).

Thus, it is naturally to expect that the global stability will be lost if the cardinality $|{{\,\mathrm{Fix}\,}}({\mathbf {F}})|\ge 2$. Note that if $m=1$ then Lemma 1 easily implies that ${{\,\mathrm{Fix}\,}}({\mathbf {F}})$ consists of exactly one point: $|{{\,\mathrm{Fix}\,}}({\mathbf {F}})|=1$. However, if $m\ge 2$, the situation is more subtle as the example below shows (see also [15]).

Example 1

First let us consider (6) with $M=m=2$,

$$\begin{aligned} (c_{ij})=\left( \begin{array}{cc} 1 &{}\quad 0 \\ 0 &{}\quad 1 \\ \end{array} \right) ,\quad (K_{ij})=\left( \begin{array}{cc} 1 &{}\quad \beta \\ \beta &{}\quad 1 \\ \end{array} \right) ,\quad \mu _j=0, \,\,S_i=S,\,\, \frac{r_i^2}{D_i\gamma _j}=:A>S \end{aligned}$$

for all $i=1,2$, where $\beta >1$ to be specified later. Then ${\mathbf {F}}=(f(v_1,v_2),f(v_2,v_1))$, with

$$\begin{aligned} f(x,y)=S-A\min \left( \frac{x^2}{(1+x)^2}, \frac{y^2}{(\beta +y)^2}\right) . \end{aligned}$$

Lemma 1 easily yields the existence of exactly one solution of (28) on the diagonal $v_1=v_2$, $0<v_1<S$. We claim that there exists yet another solution in the triangle $\varDelta =\{0<\beta v_1\le v_2<S\}$. Indeed,

$$\begin{aligned}{\mathbf {F}}|_{\varDelta }=\left( S-\frac{Av_1^2}{(1+v_1)^2}, S-\frac{Av_1^2}{(\beta +v_1)^2}\right) , \end{aligned}$$

and by Lemma 1 there exists a unique $0<{\bar{v}}_1<S$ such that $S-{\bar{v}}_1=\frac{A{\bar{v}}_1^2}{(1+{\bar{v}}_1)^2}$. Define ${\bar{v}}_2=S-\frac{A{\bar{v}}_1^2}{(\beta +{\bar{v}}_1)^2}$. Then $({\bar{v}}_1,{\bar{v}}_2)$ will be a desired fixed point if we ensure that it belongs to $\varDelta$. We have

$$\begin{aligned} \frac{ {\bar{v}}_2-\beta {\bar{v}}_1}{\beta -1}=A\biggl (g(\beta ,{\bar{v}}_1)-\frac{S}{A}\biggr ), \quad \text {where}\, g(\beta ,t)=\frac{t^2\bigl ((t+1+\beta )^2-\beta \bigr )}{(t+1)^2(t+\beta )^2} \end{aligned}$$

(30)

Notice that for any $\beta >1$, $g(\beta ,t)$ is an increasing function of $t>0$, $g(\beta ,0)=0$ and

$$\begin{aligned} \lim _{t\rightarrow \infty }g(\beta ,t)=1>\frac{S}{A}, \end{aligned}$$

therefore there exists a unique $t_\beta >0$ such that $g(\beta ,t_\beta )=\frac{S}{A}$. Next notice that $\frac{\partial g}{\partial \beta }<0$, hence $t_\beta$ is a decreasing continuous function of $\beta$. Since $\lim _{\beta \rightarrow \infty }g(\beta ,t)\equiv 1$ uniformly on any ray $(\epsilon ,\infty )$, $\epsilon >0$, we also have

$$\begin{aligned} \lim _{t\rightarrow +0}t_\beta =0. \end{aligned}$$

Therefore, there exists ${\bar{\beta }}$ such that ${\bar{v}}_1>t_{{\bar{\beta }}}$, thus $g({\bar{\beta }},{\bar{v}}_1)>g({\bar{\beta }},t_{{\bar{\beta }}})=0$ and (30) yields ${\bar{v}}_2-{\bar{\beta }}{\bar{v}}_1>0$, implying our claim. Next, since ${\bar{v}}_2>{{\bar{\beta }}} {\bar{v}}_1>{\bar{v}}_1$, the found solution is off of the diagonal. By symmetry reasons, $({\bar{v}}_2,{\bar{v}}_1)$ is also a solution of (28). Finally, since all the three solutions are distinct, the standard continuity argument implies that (28) still has three distinct solutions for $(c_{ij})=\left( \begin{array}{cc} 1 &{}\quad \epsilon _1 \\ \epsilon _1 &{}\quad 1 \\ \end{array} \right)$ and $\mu _j=\epsilon _2$ when $\epsilon _i>0$ small enough.

A careful analysis shows that for $m=2$ there always holds $|{{\,\mathrm{Fix}\,}}({\mathbf {F}})|\ge 3$ Liebig–Monod model (6). Furthermore, for any $m\ge 2$, an argument similar to Example 1 yields $|{{\,\mathrm{Fix}\,}}({\mathbf {F}})|\ge m+1$ for certain sets of parameters.

Now, let us turn to the fixed point problem (28). It is naturally to study solutions of (28) by virtue of iterations ${\mathbf {F}}^k(0)$. But (28) is non-regular in the sense that already the second iteration ${\mathbf {F}}^2(0)$ can be outside of [0, S]. Indeed, ${\mathbf {F}}^2_i(0)={\mathbf {F}}_i(S)$ becomes negative if $\gamma _j$ or $D_i$ are small enough (alternatively, $c_{ij}$ large enough).

To refine iterations, we suitably polarize (28) to get a system with the same set of fixed points. Namely, given $w\in [0,S]$ let us define ${\mathbf {V}}(w) \in [0,S]$ as the unique solution v of the system

$$\begin{aligned} S_i-v_i=\sum _{j=1}^{M} \frac{c_{ij}}{D_i} {\mathbf {X}}_j(w) \phi _j(w_1,\ldots ,w_{i-1},v_i,\ldots ,w_m),\quad i=1,\ldots ,m. \end{aligned}$$

(31)

Note that each equation of system (31) contains a single unknown variable $v_i$, thus Lemma 1 implies that for all i a unique solution $v_i$ of (31) exists and $0<v_i\le S$. Therefore, $0\ll {\mathbf {V}}(w)\le S$. Also, by the survivability condition (10) ${\mathbf {X}}_j(S)=\frac{1}{{\gamma _j}} (\phi _j(S) -\mu _j)>0$, hence

$$\begin{aligned} 0\ll {\mathbf {V}}(S)\ll S. \end{aligned}$$

(32)

Furthermore, the second part of Lemma 1 implies that ${\mathbf {V}}(w)$ is non-increasing:

$$\begin{aligned} w_1\le w_2\,\,\Rightarrow \,\,{\mathbf {V}}(w_1)\ge {\mathbf {V}}(w_2). \end{aligned}$$

Now, if v solves (28) then by the uniqueness of solution of (31) one has

$$\begin{aligned} v={\mathbf {V}}(v). \end{aligned}$$

(33)

Conversely, if v is a solution of (33) then it also solves (28). Thus, in the present setting, the fixed point problem (28) is completely equivalent to (33):

$$\begin{aligned} {{\,\mathrm{Fix}\,}}({\mathbf {F}})={{\,\mathrm{Fix}\,}}({\mathbf {V}}). \end{aligned}$$

The main advantage of ${\mathbf {V}}$ with respect to ${\mathbf {F}}$ is that by its definition,

$$\begin{aligned} {\mathbf {V}}:[0,S]\rightarrow [0,S]. \end{aligned}$$

Now, with ${\mathbf {V}}$ in hands we apply the technique of Sect. 4. Namely, using the definition (16), we see that starting with $u^0=0$, the even and odd iterations converge, respectively, to

$$\begin{aligned} \lim _{k\rightarrow \infty }{\mathbf {V}}^{2k}(0)=:\check{0}_{{\mathbf {V}}}\le {\hat{0}}_{{\mathbf {V}}}:=\lim _{k\rightarrow \infty }{\mathbf {V}}^{2k-1}(0). \end{aligned}$$

(34)

In particular,

$$\begin{aligned} \left( \check{0}_{{\mathbf {V}}},\, {\hat{0}}_{{\mathbf {V}}}\right) \in {{\,\mathrm{Fix}\,}}_2({\mathbf {V}}), \end{aligned}$$

and, furthermore, $(\check{0}_{{\mathbf {V}}},\, {\hat{0}}_{{\mathbf {V}}})$ possesses the extremal property in Proposition 2. In particular, it follows from (19) that

$$\begin{aligned} \check{0}_{{\mathbf {V}}}\le v\le {\hat{0}}_{{\mathbf {V}}},\quad \forall v\in {{\,\mathrm{Fix}\,}}({\mathbf {V}}). \end{aligned}$$

(35)

This immediately yields

Proposition 6

If the equality

$$\begin{aligned} \check{0}_{{\mathbf {V}}}={\hat{0}}_{{\mathbf {V}}} \end{aligned}$$

(36)

holds then there exists a unique special equilibrium point, i.e.

$$\begin{aligned} |{{\,\mathrm{Fix}\,}}({\mathbf {F}})|=|{{\,\mathrm{Fix}\,}}({\mathbf {V}})|=1. \end{aligned}$$

Conversely, (35) implies that the cardinality of fixed points $|{{\,\mathrm{Fix}\,}}({\mathbf {F}})|$ is an obstacle for the coincidence relation (36). Furthermore, Example 1 shows that for certain values of parameters of our system one has $|{{\,\mathrm{Fix}\,}}({\mathbf {F}})|>1$, thus, one cannot expect in general the coincidence in (36). Therefore, it is important to know when (36) holds. One such sufficient condition is presented below.

Proposition 7

Let $L_j$ be the $L^\infty$-Lipschitz constant of $\phi _j$. If $m=1$ and

$$\begin{aligned} \rho _1:=\sum _{j=1}^M\frac{\mu _j c_{1j}L_j}{D_1\gamma _j}< 1 \end{aligned}$$

(37)

or $m\ge 2$ and

$$\begin{aligned} \rho _m:=\max _{1\le i\le m} \sum _{j=1}^M\frac{(2\phi _j(S)-\mu _j)c_{ij}L_j}{D_i\gamma _j}\le 1 \end{aligned}$$

(38)

then (36) holds.

Proof

Assume by contradiction that ${\hat{0}}_{{\mathbf {V}}}=(\eta _1,\ldots ,\eta _m)>\check{0}_{{\mathbf {V}}} =(\xi _1,\ldots ,\xi _m)$ and rewrite (17) as

$$\begin{aligned} \left\{ \begin{array}{ll} S_i-\xi _i= \sum _{j=1}^M\frac{c_{ij}}{D_i\gamma _j}(\phi _j(\eta )-\mu _j)_+\phi _j(\eta ^{(i)}), \qquad \eta ^{(i)}=(\eta _1,\ldots ,\xi _i,\ldots \eta _m),\\ S_i-\eta _i= \sum _{j=1}^M\frac{c_{ij}}{D_i\gamma _j}(\phi _j(\xi )-\mu _j)_+\phi _j(\xi ^{(i)}), \qquad \,\xi ^{(i)}=(\xi _1,\ldots ,\eta _i,\ldots \xi _m) \end{array} \right. \end{aligned}$$

(39)

First let us consider the case $m=1$. Then (39) takes a simpler form

$$\begin{aligned} \left\{ \begin{array}{ll} S_1-\xi _1= \sum _{j=1}^M\frac{c_{1j}}{D_1\gamma _j} (\phi _j(\eta _1)-\mu _j)_+\phi _j(\xi _1), \\ S_1-\eta _1= \sum _{j=1}^M\frac{c_{1j}}{D_1\gamma _j} (\phi _j(\xi _1)-\mu _j)_+\phi _j(\eta _1). \end{array} \right. \end{aligned}$$

(40)

A simple analysis shows that for all $0\le a\le b$, $\mu \ge 0$, the following inequality is true

$$\begin{aligned} a(b-\mu )_+-b(a-\mu )_+=\left\{ \begin{array}{rr} 0&{}\quad \text {if}\;a\le b\le \mu \\ a(b-\mu )&{}\quad \text {if}\;a\le \mu \le b\\ \mu (b-a)&{}\quad \text {if}\;\mu \le a\le b \end{array} \right. \le \mu (b-a) \end{aligned}$$

therefore, taking into account that $\phi _j(\xi _1)\le \phi _j(\eta _1)$ and subtracting relations in (40) one obtains

$$\begin{aligned} 0<\eta _1-\xi _1\le \sum _{j=1}^M\frac{c_{1j}\mu _j}{D_1\gamma _j}(\phi _j(\eta _1)-\phi _j(\xi _1)) \le \rho _1 (\eta _1-\xi _1)< (\eta _1-\xi _1), \end{aligned}$$

a contradiction follows.

Now, let $m\ge 2$. Since $\xi ^{(i)}\ge \xi$ and $\eta ^{(i)}\le \eta$, we obtain on subtracting equations in (39) that

$$\begin{aligned} \eta _i-\xi _i\le \sum _{j=1}^M\frac{c_{ij}}{D_i\gamma _j}(f_j(\phi _j(\eta ))-f_j(\phi _j(\xi ))), \end{aligned}$$

(41)

where $f_j(x)=x(x-\mu _j)_+$ has the Lipschitz constant $(2b-\mu _j)$ on $0\le x\le b$. Combining this with the fact that $\phi _j(\eta )\le \phi _j(S)$ and $\mu _j< \phi _j(S)$ we obtain from (41) and using the definition of $\rho _m$ that

$$\begin{aligned} \Vert \eta -\xi \Vert _{\infty }< \rho _m \cdot \Vert \eta -\xi \Vert _{\infty }\le \Vert \eta -\xi \Vert _{\infty }, \end{aligned}$$

where $\Vert x\Vert _{\infty }=\max _{1\le i\le m}|x_i|$. This immediately yields the desired contradiction. $\square$

In general one has from (32) that ${\mathbf {V}}(S)\ll S$ and ${\mathbf {V}}^2(S)\le S$, hence the following simple bilateral estimates hold:

$$\begin{aligned} 0\ll {\mathbf {V}}(S)\le \check{0}_{{\mathbf {V}}} \le {\hat{0}}_{{\mathbf {V}}}\le {\mathbf {V}}^2(S)\le S. \end{aligned}$$

(42)

The latter estimate (42) is optimal in general. Indeed, if $\frac{c_{ij}}{D_i\gamma _j}$ are large enough, ${\mathbf {V}}(S)$ can be made arbitrarily small, for instance such that $\phi _j({\mathbf {V}}(S))\le \mu _j$, which yields ${\mathbf {V}}^2(S)=S$ and therefore $(\check{0}_{{\mathbf {V}}},{\hat{0}}_{{\mathbf {V}}})=({\mathbf {V}}(S), \,S)$.

Proposition 8

For any $w\in [0,S]$,

$$\begin{aligned}{}[{\mathbf {F}}(w\wedge {\mathbf {V}}(w))]_+\le {\mathbf {V}}(w)\le {\mathbf {F}}(w\vee {\mathbf {V}}(w)). \end{aligned}$$

(43)

In particular,

$$\begin{aligned}{}[{\mathbf {F}}(S)]_+\le {\mathbf {V}}(S)\le {\mathbf {V}}^2(S)\le {\mathbf {F}}([{\mathbf {F}}(S)]_+). \end{aligned}$$

(44)

Here $x\vee y$ (resp. $x\wedge y$) denote the vector whose ith coordinate is $\min (x_i,y_i)$ (resp. $\max (x_i,y_i)$).

Proof

Let $v={\mathbf {V}}(w)$. Since for any i, $w\vee v\le (w_1,\ldots ,w_{i-1},v_i,\ldots ,w_m)$, it follows from the monotonicity of ${\mathbf {F}}$ and (31) that

$$\begin{aligned} {\mathbf {F}}_i(w\vee v)&\ge {\mathbf {F}}_i(w_1,\ldots ,w_{i-1},v_i,\ldots ,w_m)\\&=S_i-\sum _{j=1}^{M} \frac{c_{ij}}{D_i} {\mathbf {X}}_j(w) \phi _j(w_1,\ldots ,w_{i-1},v_i,\ldots ,w_m)\\&=v_i={\mathbf {V}}_i(w), \end{aligned}$$

which yields the right inequality in (43). The left one follows by a similar argument from $w\wedge v\ge (w_1,\ldots ,w_{i-1},v_i,\ldots ,w_m)$ and the fact that ${\mathbf {V}}(w)\ge 0$. Then (44) follows from $0\ll {\mathbf {V}}(S)\le {\mathbf {V}}^2(S)\le S$ and (43). $\square$

7 Bilateral estimates

As it was pointed out before, Example 1 shows that a priori the asymptotic behaviour of solutions to (1)–(2) can be rather complicated for $m\ge 2$. On the other hand, the result below shows that the global dynamics is completely controlled by the finite-dimensional fixed point problem (28) and the characteristic parameters in (17).

Theorem 1

Let (x(t), v(t)) be the solution of (1), (2), (3). Then in notation of Sects. 4 and 5:

$$\begin{aligned} \check{0}_{{\mathbf {V}}}\le \lim \inf _{t\rightarrow \infty }v(t)&\le \lim \sup _{t\rightarrow \infty }v(t)\le {\hat{0}}_{{\mathbf {V}}}, \end{aligned}$$

(45)

$$\begin{aligned} {\mathbf {X}}(\check{0}_{{\mathbf {V}}})\le \lim \inf _{t\rightarrow \infty }x(t)&\le \lim \sup _{t\rightarrow \infty }x(t)\le {\mathbf {X}}({\hat{0}}_{{\mathbf {V}}}). \end{aligned}$$

(46)

In particular, if $\check{0}_{{\mathbf {V}}}={\hat{0}}_{{\mathbf {V}}}$ then all solutions of (1)–(2) converge to a unique special equilibrium point.

Proof

As the first step, we reformulate the original system as an appropriate integral equation for unknown function v(t). Let $w(t):[0,\infty )\rightarrow [0,S]$ be a continuous vector function with $w(0)=v(0)$ and having a limit $\lim _{t\rightarrow \infty }w(t)={\bar{w}}$. Then

$$\begin{aligned} {\mathscr {X}}_i(w)(t)=x_i(0)\left( e^{- \int _0^t (\phi _j(w(s))-\mu _j) ds} + \gamma _j x_i(0) \int _0^t e^{- \int _{t_1}^t (\phi _j(w(s))-\mu _j) ds} dt_1\right) ^{-1}, \end{aligned}$$

(47)

solves (1) with v(t) replaced by w(t). Clearly, ${\mathscr {X}}(w)(t)$ is a non-decreasing function of w, ${\mathscr {X}}(w)(0)=x(0)$ and one can readily verify that

$$\begin{aligned} \lim _{t\rightarrow \infty }{\mathscr {X}}_i(w)(t)=\frac{1}{{\gamma _j}} (\phi _j({\bar{w}}) -\mu _j)_{+}= {\mathbf {X}}_i({\bar{w}}). \end{aligned}$$

(48)

Next, let ${\mathscr {V}}(w)(t)$ denote the solution u(t) of the system below [obtained from (2) with x(t) replaced by (47)):

$$\begin{aligned} \begin{aligned} \frac{\mathrm{d}u_i}{\mathrm{d}t}&=D_i(S_i -u_i) - \sum _{i=1}^M c_{ ij} \; \phi _j(w_1,\ldots ,w_{i-1},u_i,\ldots ,w_m){\mathscr {X}}_j(w)(t)\\ u_i(0)&=v_i(0), \qquad i=1,\dots , m. \end{aligned} \end{aligned}$$

(49)

Let $C^1_S[0,T]$ denote the set of $C^1$ vector functions u(t) on [0, T] such that $0\le u(t)\le S$. Then

$$\begin{aligned} {\mathscr {V}}:C^1_S[0,T]\rightarrow C^1_S[0,T], \quad \forall T>0. \end{aligned}$$

(50)

Next, note that ${\mathscr {V}}(w)$ is a non-increasing functional of w. Indeed, let $0\le w(t)\le {\widetilde{w}}(t)$ for all $t\ge 0$, and let $u_i(t)$ and ${\widetilde{u}}_i(t)$ be the corresponding solutions of (49). Denote by $F_i(u_i(t),t)$ and ${\widetilde{F}}_i({\widetilde{u}}_i(t),t)$ the right-hand side of (49) corresponding to w(t) and ${\widetilde{w}}(t)$, respectively. Then the $F_i(z,t)$ and ${\widetilde{F}}_i(z,t)$ satisfy the conditions of Lemma 2 and $u_i(0)={\widetilde{u}}_i(0)$, therefore $u_i(t)\ge {\widetilde{u}}_i(t)$ for all t, as desired.

Furthermore, we claim that

$$\begin{aligned} \lim _{t\rightarrow \infty }{\mathscr {V}}(w)(t)= {\mathbf {V}}({\bar{w}}). \end{aligned}$$

(51)

Indeed, rewrite (49) as $u_i'(t)=F_i(u_i(t),t)$, where

$$\begin{aligned} F_i(z,t):=D_i(S_i -z) - \sum _{j=1}^M c_{ ij} \; \phi _j(w_1(t),\ldots ,z,\ldots ,w_m(t)){\mathscr {X}}_j(w)(t). \end{aligned}$$

Then $F_i(z,t)$ obviously satisfies conditions (a) and (b) of Lemma 3 with $c=D_i$. To verify (c), note that by (48) for any $z\in [0,S]$:

$$\begin{aligned} {\bar{F}}_i(z):=\lim _{t\rightarrow \infty } F_i(z,t)=D_i(S_i -z) - \sum _{j=1}^M c_{ ij} \; \phi _j({\bar{w}}_1,\ldots ,z,\ldots ,{\bar{w}}_m){\mathbf {X}}_j({\bar{w}}). \end{aligned}$$

Comparing the latter expression with (31), we conclude that $z={\mathbf {V}}_i({\bar{w}})$ is the unique root of ${\bar{F}}_i(z)=0$ in [0, S]. Now, let $0\le z_i(t)<S$ be the unique solution of $F_i(z_i(t),t)=0$, $t\ge 0$. Suppose that $t_k\nearrow \infty$ realizes ${\bar{z}}:=\lim \sup _{t\rightarrow \infty } z_i(t)$. Then

$$\begin{aligned} 0=\lim _{k\rightarrow \infty }F_i(z_i(t_k),t_k)={\bar{F}}_i({\bar{z}}) \quad \Rightarrow \quad {\bar{z}}={\mathbf {V}}_i({\bar{w}}). \end{aligned}$$

Similarly one shows that ${\mathbf {V}}_i({\bar{w}})=\lim \inf _{t\rightarrow \infty } z_i(t)$. Thus, $\lim _{t\rightarrow \infty } z_i(t)={\mathbf {V}}_i({\bar{w}})$ exists, as desired. Applying Lemma 3 yields (51).

In the present setting, if (x(t), v(t)) is the solution of (1), (2), (3) then $v=v(t)$ satisfies the fixed point problem

$$\begin{aligned} v={\mathscr {V}}(v), \end{aligned}$$

(52)

then $x=x(t)$ is recovered by

$$\begin{aligned} x={\mathscr {X}}(v). \end{aligned}$$

(53)

Now we show that v(t) can be obtained as the limit of iterations

$$\begin{aligned} v^{k}(t)={\mathscr {V}}^k(v^0)(t), \quad k\ge 1, \quad \text {where }v^0(t)\equiv 0. \end{aligned}$$

As ${\mathscr {V}}$ is non-increasing and ${\mathscr {V}}(v)=v$, one has

$$\begin{aligned} 0=v^0\le v\le v^{1}\le S. \end{aligned}$$

Since ${\mathscr {V}}^2$ is non-decreasing and by (50) $v^2\ge 0=v^0$, one readily obtains

$$\begin{aligned} v^0\le v^2\le \cdots v^{2k}\le \cdots v\le \cdots \le v^{2k-1}\le \cdots \le v^3\le v^1 \end{aligned}$$

and $v^k\in C^1_S[0,\infty )$. For any fixed $T>0$, the operator ${\mathscr {V}}:C^1_S[0,T]\rightarrow C^1_S[0,T]$ is compact, hence both the odd $v^{2k-1}(t)$ and even $v^{2k}(t)$ terms converge in $C^1[0,T]$, therefore the following limits are well-defined for any $t\ge 0$:

$$\begin{aligned} \check{v}(t)=\lim _{k\rightarrow \infty }v^{2k}(t),\;\;\; {\hat{v}}(t)=\lim _{k\rightarrow \infty }v^{2k+1}(t), \end{aligned}$$

(54)

and

$$\begin{aligned} {\mathscr {V}}(\check{v})={\hat{v}}\quad {\mathscr {V}}({\hat{v}})=\check{v}. \end{aligned}$$

(55)

Since $v^0=0\le v$ we also have by (52) that $v^{2k}\le v\le v^{2k+1}$, thus implying

$$\begin{aligned} {\check{v}}(t)\le v(t)\le {{\hat{v}}}(t), \qquad {\check{x}}(t)\le x(t)\le {{\hat{x}}}(t), \end{aligned}$$

(56)

where ${\hat{x}}={\mathscr {X}}({\hat{v}})$, $\check{x}={\mathscr {X}}(\check{v})$, and $(\check{x},{\hat{v}})$ and $({\hat{x}},\check{v})$ solve, respectively,

$$\begin{aligned} (\check{x},{\hat{v}}):&\quad \frac{\mathrm{d}\check{x}_j}{\mathrm{d}t}=\check{x}_j ( \phi _j(\check{v})- \mu _j - \gamma _{j} \check{x}_j), \quad \frac{\mathrm{d}{\hat{v}}_i}{\mathrm{d}t}=D_i(S_i -{\hat{v}}_i) - \sum _{j=1}^M c_{ij} \; \check{x}_j \; \phi _j({\hat{v}}),\\ ({\hat{x}},\check{v}):&\quad \frac{\mathrm{d}{\hat{x}}_j}{\mathrm{d}t}={\hat{x}}_j ( \phi _j({\hat{v}})- \mu _j - \gamma _{j} {\hat{x}}_j), \quad \frac{\mathrm{d}\check{v}_i}{\mathrm{d}t}=D_i(S_i -\check{v}_i) - \sum _{j=1}^M c_{ij} \; {\hat{x}}_j \; \phi _j(\check{v}). \end{aligned}$$

Taking the difference yields that $(\xi ,\eta ):=({\hat{x}}-\check{x},{\hat{v}}-\check{v})$ satisfies a homogeneous system of ODEs with bounded coefficients (recall that $\phi _j$ are Lipschitz). Since $(\xi ,\eta )$ has the zero Cauchy data, we conclude by uniqueness for the Cauchy problem and (56) that $\check{v}(t)= {\hat{v}}(t)=v(t)$ and $\check{x}(t)= {\hat{x}}(t)=x(t)$. In summary, for any fixed $t>0$ one has

$$\begin{aligned} x(t)=\lim _{k\rightarrow \infty } x^k(t),\qquad v(t)=\lim _{k\rightarrow \infty } v^k(t), \end{aligned}$$

(57)

where by (51) ${\bar{v}}^k:=\lim _{t\rightarrow \infty } v^k(t)={\mathbf {V}}^k(0)$, ${\bar{v}}^0=0.$ Applying the results of Sect. 4 to (54) yields $\lim _{k\rightarrow \infty } {\bar{v}}^{2k}=\check{0}_{{\mathbf {V}}},$$\lim _{k\rightarrow \infty } {\bar{v}}^{2k-1}={\hat{0}}_{{\mathbf {V}}},$ which proves (45). Similarly, (46) follows from (48) and (57). $\square$

Combining the obtained estimates with Proposition 7 implies the following global stability result.

Proposition 9

(Global stability) If $m=1$ and (37) holds or $m\ge 2$ and (38) holds then (1)–(2) is globally stable: any solution with Cauchy data (3) converges to a unique equilibrium point $\check{0}_{{\mathbf {V}}}={\hat{0}}_{{\mathbf {V}}}$.

Numerical simulations in [13] show that certain solutions of the standard model with $\gamma _j=0$ and $m\ge 3$ have periodic (chaotic) dynamics. Proposition 9 shows that if the self-limitation constants $\gamma _j$ or dilution rates $D_i$ are large enough, the global behaviour of the modified model becomes stable for any choice of m and M.

In fact, one can choose the parameters of the system such that the strong persistence holds, see the corollary below. To present our result, we need to define an analogue of the lowest break-even concentration $R^*$ in (7) for general response functions $\phi _j$. Let us consider the set

$$\begin{aligned} {\mathscr {R}}:=\{v\in [0,S]: \phi _j(v)> \mu _j \,\,\, \text {for all }j\}. \end{aligned}$$

(58)

Note that by (10), ${\mathscr {R}}\ne \emptyset$.

Proposition 10

(Strong persistence) In notation of Proposition 9, there exists $\rho _0=\rho _0({\mathscr {R}})>0$ such that if $\rho \le \rho _0$ then any solution of (1)–(2)–(3) converges to a unique equilibrium point with

$$\begin{aligned} \lim _{t_\rightarrow \infty }x_i(t)>0\quad \text { for all }1\le i\le M. \end{aligned}$$

Proof

By (10), $S\in {\mathscr {R}}$, therefore the number

$$\begin{aligned} \delta :=\sup \{t\ge 0: (S_1-t,\ldots ,S_m-t)\in {\mathscr {R}}\} \end{aligned}$$

is well-defined and positive. Since $\delta \le \Vert S\Vert _\infty$, we have $\rho _0:=\delta /\Vert S\Vert _{\infty }\le 1$. If $\rho \le \rho _0$ then by Proposition 9 any solution with Cauchy data (3) converges to a unique equilibrium point $0\ll \xi \ll S$ satisfying (28). We have for all $1\le i\le m$

$$\begin{aligned} \begin{aligned} S_i-\xi _i&=\sum _{j=1}^M \frac{c_{ij}}{D_i\gamma _j}(\phi _j(\xi )-\mu _j)_+\phi _j(\xi )\equiv \sum _{j=1}^M \frac{c_{ij}}{D_i\gamma _j}[f_j(\phi _j(\xi ))-f_j(0)]\\&< \rho \Vert \xi \Vert _\infty \le \frac{\delta \Vert \xi \Vert _\infty }{\Vert S\Vert _{\infty }}<\delta \end{aligned} \end{aligned}$$

(59)

Therefore, $\xi \in {\mathscr {R}}$, implying by (58) and (29) that $\lim _{t\rightarrow \infty }x_j(t)={\mathbf {X}}_j(\xi )>0$ for all j, as desired. $\square$

In general, one has from (45), (42) and (44) the following explicit a priori estimate.

Corollary 1

Let (x(t), v(t)) be the solution of (1), (2), (3) and let the survivability condition (10) holds. Then

$$\begin{aligned} {[}{\mathbf {F}}(S)]_+\le \lim \inf _{t\rightarrow \infty }v(t)&\le \lim \sup _{t\rightarrow \infty }v(t)\le {\mathbf {F}}([{\mathbf {F}}(S)]_+)\\ {\mathbf {X}}([{\mathbf {F}}(S)]_+)\le \lim \inf _{t\rightarrow \infty }x(t)&\le \lim \sup _{t\rightarrow \infty }x(t)\le {\mathbf {X}}({\mathbf {F}}([{\mathbf {F}}(S)]_+)). \end{aligned}$$

where ${\mathbf {F}}_i(S)=S_i-\sum _{j=1}^M\frac{c_{ij}}{D_i\gamma _j} (\phi _j(S)-\mu _j)\phi _j(S)$, $1\le i\le m$ and ${\mathbf {X}}$ is defined by (29).

8 Discussion

In this paper, we established sufficient conditions for the global stability and persistence of a chemostat-like model with self-limitations. For the Liebig-Mondoc model (6), one has $L_j= r_j/\min _{i}\{K_{ij}\}$ and $\phi _j(S)\le r_j$. It is interesting to compare our result with simulations in [13] rigorously proved in [19], see especially Section 5 there. In that example, Huisman and Weissing assume in the present notation that $m=M=3$, $S_j=10$, $r_j=1$, $D_j=0.25$ for all three species and matrices $K_{ij}$ and $c_{ij}$ be chosen as in [19, p. 38]. Then if $\gamma _j=0$ then Theorem 3.1 in [19] implies the existence of a nontrivial periodic oscillation. On the other hand, it follows from (38) that if $\gamma _j\ge 1.64$, $j=1,2,3$, then any solution is global stable, for arbitrary positive initial data. In general, given arbitrary data, (38) explicitly defines the critical values $\gamma _j^*$ such that the system is globally stable for $\gamma _j>\gamma _j^*$.