# Global stability and persistence of complex foodwebs

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## 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.

## Keywords

Global stability Persistence Period-two-points Non-increasing maps Complex foodwebs Self-limitation Multiple resources## Mathematics Subject Classification

34D20 92D25 47H10 34C60## 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

*i*; \({\mathbb R}_{+}^n\) denotes the nonnegative cone \( \{x\in {\mathbb R}^n:x\ge 0\}\) and for \(a\le b\), \(a,b\in {\mathbb R}^n\)

*a*and

*b*.

*M*species competing for

*m*complementary resources is governed by chemostat-like equations

*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:

*j*and \(K_{ij}\) is the half-saturation constant for resource

*i*of species

*j*. Obviously, the functions (6) meet the above conditions.

*lowest break-even concentration*

*k*then

## 3 Preliminaries

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

### Proposition 1

*x*(

*t*),

*v*(

*t*)) of (1), (2), (3) is well-defined and bounded for all \(t\ge 0\) and

*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

*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

*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 \(\lim _{t\rightarrow \infty }x_j(t)=0\). \(\square \)

*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:

*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

### 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

*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

### Lemma 3

*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 \(\lim \limits _{t\rightarrow \infty }z(t)={\bar{z}}\).

### Proof

*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

*u*(

*t*) is non-decreasing, therefore there exists

*u*(

*t*) and (11) implies

*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

*a*,

*b*), \(a,b\in D\), is called a

*period-two-point*[7, p. 387], or \((a,b)\in {{\,\mathrm{Fix}\,}}_2(G)\), if

*c*,

*c*).

*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:

*G*). Hence, it follows by induction that

*G*:

### Proposition 2

*G*.

### Proof

*G*is a non-increasing, one has

*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

### Proof

*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)

## 5 Stratification of equilibrium points

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

*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 \(E_{\emptyset }\):

### Proposition 4

### 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, \(v\ll S\). \(\square \)

*v*is determined uniquely by

*v*solves the fixed point problem

*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).

### Proposition 5

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

### Proof

*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

*E*, we shall distinguish the

*special*ones, namely those contained in

*x*,

*v*) is said to be a special (equilibrium) point if and only if

*v*is a solution of the fixed point problem

*x*is given by

*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

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).

*v*of the system

*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

*non-increasing*:

*v*solves (28) then by the uniqueness of solution of (31) one has

*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):

### Proposition 6

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

### Proof

### Proposition 8

*i*th coordinate is \(\min (x_i,y_i)\) (resp. \(\max (x_i,y_i)\)).

### Proof

*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

## 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

*x*(

*t*),

*v*(

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

### Proof

*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

*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

*u*(

*t*) of the system below [obtained from (2) with

*x*(

*t*) replaced by (47)):

*u*(

*t*) on [0,

*T*] such that \(0\le u(t)\le S\). Then

*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.

*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

*x*(

*t*),

*v*(

*t*)) is the solution of (1), (2), (3) then \(v=v(t)\) satisfies the fixed point problem

*v*(

*t*) can be obtained as the limit of iterations

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*.

### Proposition 10

### Proof

*j*, as desired. \(\square \)

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

### Corollary 1

*x*(

*t*),

*v*(

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

## 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 \(\gamma _j>\gamma _j^*\).

## Notes

### Acknowledgements

The authors express their gratitude to the editor and the anonymous reviewers for valuable and constructive comments.

## References

- 1.Allesina, S.: Ecology: the more the merrier. Nature
**487**(7406), 175–176 (2012)CrossRefGoogle Scholar - 2.Allesina, S., Pascual, M.: Network structure, predator-prey modules, and stability in large food webs. Theor. Ecol.
**1**(1), 55–64 (2008)CrossRefGoogle Scholar - 3.Allesina, S., Tang, S.: Stability criteria for complex ecosystems. Nature
**483**(7388), 205–208 (2012)CrossRefGoogle Scholar - 4.Armstrong, R., McGehee, R.: Competitive exclusion. Am. Nat.
**115**(2), 151–170 (1980)MathSciNetCrossRefGoogle Scholar - 5.Clarke, F.H.: On the inverse function theorem. Pac. J. Math.
**64**(1), 97–102 (1976)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Haddad, W., Chellaboina, V., Hui, Q.: Nonnegative and Compartmental Dynamical Systems. Princeton University Press, Princeton (2010)CrossRefzbMATHGoogle Scholar
- 7.Hale, J., Lunel, V.S.: Introduction to Functional-Differential Equations. Applied Mathematical Sciences, vol. 99. Springer, New York (1993)CrossRefzbMATHGoogle Scholar
- 8.Hardin, G.: The competitive exclusion principle. Science
**131**(3409), 1292–1297 (1960)CrossRefGoogle Scholar - 9.Hsu, S.: Limiting behavior for competing species. SIAM J. Appl. Math.
**34**(4), 760–763 (1978)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Hsu, S.: A survey of constructing Lyapunov functions for mathematical models in population biology. Taiwan. J. Math.
**9**(2), 151–173 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Hsu, S., Hubbell, S., Waltman, P.: A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms. SIAM J. Appl. Math.
**32**(2), 366–383 (1977)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Huisman, J., Weissing, F.: Biological conditions for oscillations and chaos generated by multispecies competition. Ecology
**82**(10), 2682–2695 (2001)CrossRefGoogle Scholar - 13.Huisman, J., Weissing, F.: Biodiversity of plankton by species oscillations and chaos. Nature
**402**(6760), 407–410 (1999)CrossRefGoogle Scholar - 14.Kozlov, V., Tkachev, V., Vakulenko, S., Wennergren, U.: Biodiversity and robustness of large ecosystems. Ecol. Complex.
**36**, 101–109 (2018)CrossRefGoogle Scholar - 15.Kozlov, V., Vakulenko, S.: On chaos in Lotka–Volterra systems: an analytical approach. Nonlinearity
**26**(8), 2299–2314 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 16.Kozlov, V., Vakulenko, S., Wennergren, U.: Stability of ecosystems under invasions. Bull. Math. Biol.
**78**(11), 2186–2211 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Kozlov, V., Vakulenko, S., Wennergren, U.: Biodiversity, extinctions, and evolution of ecosystems with shared resources. Phys. Rev. E
**95**, 032413 (2017)CrossRefGoogle Scholar - 18.de Leenheer, P., Li, B., Smith, H.: Competition in the chemostat: some remarks. Can. Appl. Math. Q.
**11**(3), 229–248 (2003)MathSciNetzbMATHGoogle Scholar - 19.Li, B.: Periodic coexistence in the chemostat with three species competing for three essential resources. Math. Biosci.
**174**(1), 27–40 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Li, B., Smith, H.: How many species can two essential resources support? SIAM J. Appl. Math.
**62**(1), 336–366 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 21.Roy, S., Chattopadhyay, J.: Towards a resolution of ‘the paradox of the plankton’: a brief overview of the proposed mechanisms. Ecol. Complex.
**4**(1), 26–33 (2007)CrossRefGoogle Scholar - 22.Smith, H., Li, B.: Competition for essential resources: a brief review. In: Ruan, S., Wolkowicz, G.S., Wu, J. (eds.) Dynamical Systems and Their Applications in Biology (Cape Breton Island, NS, 2001). Fields Institute Communications, vol. 36, pp. 213–227. American Mathematical Society, Providence (2003)Google Scholar
- 23.Tilman, D.: Resources: a graphical-mechanistic approach to competition and predation. Am. Nat.
**116**(3), 362–393 (1980)CrossRefGoogle Scholar - 24.Volterra, V.: Leçons sur la théorie mathématique de la lutte pour la vie. Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics]. Éditions Jacques Gabay, Sceaux (1990). Reprint of the 1931 originalGoogle Scholar

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