Global dynamics of a mutualism–competition model with one resource and multiple consumers

Abstract

Recent simulation modeling has shown that species can coevolve toward clusters of coexisting consumers exploiting the same limiting resource or resources, with nearly identical ratios of coefficients related to growth and mortality. This paper provides a mathematical basis for such as situation; a full analysis of the global dynamics of a new model for such a class of n-dimensional consumer–resource system, in which a set of consumers with identical growth to mortality ratios compete for the same resource and in which each consumer is mutualistic with the resource. First, we study the system of one resource and two consumers. By theoretical analysis, we demonstrate the expected result that competitive exclusion of one of the consumers can occur when the growth to mortality ratios differ. However, when these ratios are identical, the outcomes are complex. Either equilibrium coexistence or mutual extinction can occur, depending on initial conditions. When there is coexistence, interaction outcomes between the consumers can transition between effective mutualism, parasitism, competition, amensalism and neutralism. We generalize to the global dynamics of a system of one resource and multiple consumers. Changes in one factor, either a parameter or initial density, can determine whether all of the consumers either coexist or go to extinction together. New results are presented showing that multiple competing consumers can coexist on a single resource when they have coevolved toward identical growth to mortality ratios. This coexistence can occur because of feedbacks created by all of the consumers providing a mutualistic service to the resource. This is biologically relevant to the persistence of pollination–mutualisms.

Appendices

Appendix A. Proof of Proposition 2.1

Proof

Let $N(t)= (N_1(t), N_2(t), N_3(t))$ be a solution of (2.3) with a fixed initial value $N(0) \ge 0$. It follows from the first equation of (2.3) that

$$\begin{aligned} \frac{dN_1}{dt} \le N_1 \left( \bar{r}_1 - d_1N_1\right) . \end{aligned}$$

Let $K_1 = \bar{r}_1 / d_1$. The comparison principle (Hale 1969) implies $\limsup _{t \rightarrow \infty }N_1(t) \le K_1.$ Then for $\delta _0 >0$ small, there is $T_1 >0$ such that if $t > T_1$, then $N_1(t) \le K_1 + \delta _0$.

It follows from the second equation of (2.3) that if $t> T_1$, then

$$\begin{aligned} \frac{dN_2}{dt}= & {} \frac{N_2 }{1 + N_2 + N_3} [ -r_2 (1 + N_2 + N_3) + a_{21}N_1 ] \\\le & {} \frac{N_2 }{1 + N_2 + N_3} [ -r_2 (1 + N_2 + N_3) + a_{21} (K_1 + \delta _0) ] \\\le & {} \frac{N_2 }{1 + N_2 + N_3} [ a_{21} (K_1 + \delta _0) -r_2 -r_2 N_2 ) ]. \end{aligned}$$

Let $K_2 = [ a_{21} (K_1 + \delta _0) -r_2] / r_2$. If $K_2 \le 0$, then $\lim _{t \rightarrow \infty } N_2(t) =0$, which implies that there is $T_{21}> T_1 >0$ such that when $t>T_{21}$, we have $N_2(t) < \delta _0$. If $K_2 > 0$, then

$$\begin{aligned} \frac{dN_2}{dt} \le \frac{N_2 }{r_2 (1 + N_2 + N_3)} (K_2 - N_2 ) <0 \quad {\text {as}}\quad N_2 > K_2 \end{aligned}$$

which implies $\limsup _{t \rightarrow \infty } N_2(t) \le K_2$. Then there is $T_{22}> T_1 >0$ such that when $t>T_{22}$, we have $N_2(t) < K_2 + \delta _0$. Thus, there exists $T_2 >0$ such that if $t>T_2$, then $N_2(t) < |K_2| + \delta _0$.

Let $K_3 = [ a_{31} (K_1 + \delta _0) -r_3] / r_3$. By the third equation of (2.3) and a discussion similar to that for $K_2$, we obtain that there is $T_3 >T_2$ such that if $t > T_3$, then $N_3(t) < |K_3| + \delta _0$.

Let $T = T_3$. If $t > T$, then $||N(t)|| = \sum _{i=1}^3 N_i(t) \le \sum _{i=1}^3 M_i$ with $M_i= |K_i|+ \delta _0$. Thus solutions of (2.3) are bounded. $\square$

Appendix B. Proof of two cases in Theorem 3.3

Proof

(ii) The case of $\lambda _1^{(2)} = 0$.

From $\lambda _1^{(2)} = 0$, we have $a_{21} = a_{21}^*$. If $a_{21}^* < a_{21}^1$, there is no positive equilibrium in system (3.1) by Proposition 3.2. If $a_{21}^* = a_{21}^1$, we have $b_{12} = \bar{r}_1/2$ and $a_{21} = a_{21}^0$, which implies that $\Delta = 0$ and there is no positive equilibrium in system (3.1). Thus, $E_1$ is globally asymptotically stable.

If $a_{21} > a_{21}^1$, $E_{12}^+$ is the unique positive equilibrium by Proposition 3.2. We apply the central manifold theorem to show that $E_1$ has no stable manifold in int$R_+^2$, which implies that $E_{12}^+$ is globally asymptotically stable. The following transformation can change system (3.1) into a standard form near $E_1$:

$$\begin{aligned} \left( \begin{array}{lcr} x_1 \\ x_2 \end{array} \right) = \left( \begin{array}{lcr} N_1 - \bar{N}_1 \\ N_2 \end{array} \right) , \quad \left( \begin{array}{lcr} y_1 \\ y_2 \end{array} \right) = \frac{1}{d_1 } \left( \begin{array}{l@{\quad }c@{\quad }c@{\quad }r} d_1 &{} - b_{12}\\ 0 &{}1 \end{array} \right) \left( \begin{array}{lcr} x_1 \\ x_2 \end{array} \right) . \end{aligned}$$

Then system (3.1) can be written as

$$\begin{aligned} \frac{dy_1}{dt}&= \lambda _1^{(1)} y_1 -d_1 \left( y_1 + b_{12} y_2\right) ^2 + b_{12}d_1 \left( y_1 + b_{12} y_2\right) y_2 \nonumber \\&\quad - b_{12}d_1^2 \bar{N}_1 y_2^2 - \frac{ b_{12}r_2 y_2 }{ \bar{N}_1} \left[ y_1 + y_2 \left( 2 b_{12} - \bar{r}_1 \right) \right] +o\left( |y|^2\right) \nonumber \\ \frac{dy_2}{dt}&= r_2 y_2 \left[ -1+ \frac{\left( y_1 + b_{12} y_2 + \bar{N}_1\right) d_1 }{\bar{N}_1 (d_1+ y_2 )} \right] \nonumber \\&= \frac{r_2 y_2 }{ \bar{N}_1} \left[ y_1 + y_2 \left( 2 b_{12} - \bar{r}_1 \right) \right] +o\left( |y|^2\right) \end{aligned}$$

(6.1)

which implies that the solution $y_2 =0$ is a stable manifold of equilibrium (0, 0) in system (6.1). Let $y_1 = \phi (y_2) =a y_2^2 + o(y_2^2)$ be the central manifold of (6.1) at (0, 0). From $a_{21} =a_{21}^* > a_{21}^1$, we have $b_{12} = \hat{r}_1 - r_2 d_1 /a_{21} > \hat{r}_1 /2.$ A long but straightforward computation shows that

$$\begin{aligned} a= \frac{ b_{12} }{\lambda _1^{(1)} \bar{N}_1} \left[ d_1^2 \bar{N}_1^2 + r_2 \left( 2 b_{12} - \bar{r}_1 \right) \right] <0. \end{aligned}$$

Thus equilibrium (0, 0) is a saddle-node point. On the central manifold $y_1 = \phi (y_2)$, we have $dy_2/dt >0$. Then equilibrium (0, 0) has no stable manifold in int$R_+^2$. Thus, $E_1$ has no stable manifold in int$R_+^2$ and $E_{12}^+$ is globally asymptotically stable.

(iii) The case of $\lambda _1^{(2)} < 0, \Delta =0$ and $a_{21} > a_{21}^1$.

From $\lambda _1^{(2)} < 0$, we have $a_{21} < a_{21}^*$ and equilibrium $E_1$ is locally asymptotically stable.

Since $\Delta = 0$ and $a_{21} > a_{21}^1$, the two positive equilibria $E_{12}^-$ and $E_{12}^+$ coincide by Proposition 3.2. By (3.2), the Jacobian matrix of (3.1) at a positive equilibrium $(N_1,N_2)$ is

$$\begin{aligned} J = \left( \begin{array}{l@{\quad }c@{\quad }c@{\quad }r} - d_1N_1 &{} b_{12} N_1 g^2 \\ a_{21}N_2 g &{}-a_{21} N_1N_2 g^2 \end{array} \right) , \end{aligned}$$

(6.2)

which implies tr$J(E_{12}^\pm ) <0$. A direct computation shows that

$$\begin{aligned} \det J(E_{12}^\pm ) = \pm N_1N_2 g^2 \sqrt{ \Delta } =0, \end{aligned}$$

(6.3)

which implies that there is a simple zero eigenvalue of $J(E_{12}^-)$. We apply Sotomayor’s theorem Perko (2001) to show that saddle-node bifurcation occurs at $E_{12}^-$ when $\Delta =0$. For the simple zero eigenvalue, we have the left and right eigenvectors of $J(E_{12}^-)$ by (6.2):

$$\begin{aligned} w = (a_{21}N_2 g, d_1N_1 )^T,\quad v = \left( b_{12} g^2, d_1 \right) ^T. \end{aligned}$$

Let $\mu = a_{21}$ be the parameter in Sotomayor’s theorem. Let $F=(F_1,F_2)^T$ be the righthand side of (3.1). Then we have $F_\mu = (0, N_1N_2g ) ^T.$ Thus $w^T F_\mu = d_1N_1^2N_2g>0$ at $E_{12}^-$ and $\mu =a_{21}^0$. Direct computations show that

$$\begin{aligned} \frac{\partial ^2 F_1 }{\partial N_1^2}= & {} - 2 d_1,\quad \frac{\partial ^2 F_1 }{\partial N_1 \partial N_2} = b_{12} g^2,\quad \frac{\partial ^2 F_1 }{\partial N_2^2} = -2 b_{12} N_1 g^3, \\ \frac{\partial ^2 F_2 }{\partial N_1^2}= & {} 0,\quad \frac{\partial ^2 F_2 }{\partial N_1 N_2} = a_{21} g [1-N_2 g ],\quad \frac{\partial ^2 F_2 }{\partial N_2^2} = 2 r_2 g ( -1+ N_2 g ). \end{aligned}$$

A long but straightforward computation shows that

$$\begin{aligned} w \cdot D^2F(v,v) = - 2 d_1^2 a_{21} b_{12} N_1 N_2 g^4 <0. \end{aligned}$$

Thus, the Sotomayor’s theorem implies that saddle-node bifurcation occurs at $E_{12}^-$ when $a_{21} = a_{21}^0$: (1) If $a_{21} > a_{21}^0$, then $\det J (E_{12}^+) >0$ and $\det J (E_{12}^-) <0$, which implies that $E_{12}^+$ is a stable node and $E_{12}^-$ is a saddle point. (2) If $a_{21} = a_{21}^0$, $E_{12}^-$ is a saddle-node point. (3) If $a_{21} < a_{21}^0$, there is no positive equilibrium and $E_1$ is globally asymptotically stable.

It follows from Proposition 3.1 that when $E_{12}^-$ and $E_{12}^+$ exist, the separatrices of $E_{12}^-$ subdivide the interior of $(N_1,N_2)$-plane into two regions: one is the basin of attraction of $E_1$, while the other is that of $E_{12}^+$. Thus, the result in the second case is proven. $\square$

Appendix C. Proof of Theorem 4.4

Proof

Since the proof for (ii) is similar to that for (ii) in Theorem 3.3, we omit the details.

• (i) Since $\lambda _1^{(2)} > 0$, equilibrium $E_1$ is a saddle point and has no stable manifold in int$R_+^2$. From $\lambda _1^{(2)} > 0$ we have $a_{21} > a_{21}^*$ and $a_{21} > a_{21}^0$. By Proposition 4.3, $\hat{E}_{12}^+$ is the unique positive equilibrium. By Proposition 4.2, $\hat{E}_{12}^+$ is globally asymptotically stable.

• (ii) It follows from $\lambda _1^{(2)} < 0$ that the equilibrium $E_1$ is asymptotically stable. From $\lambda _1^{(2)}<0$ we obtain $a_{21} < a_{21}^*$. When $a_{21} \le a_{21}^1$, there is no positive equilibrium in system (4.1) by Proposition 4.3. Thus $E_1$ is globally asymptotically stable.

Assume $a_{21} > a_{21}^1$. If $\Delta \ge 0$, then $a_{21} \ge a_{21}^0$ and there are two positive equilibria $\hat{E}_{12}^-$ and $\hat{E}_{12}^+$ by Proposition 4.3. Let $\hat{g}(N_2) = 1/(1 + N_2 + c N_2^s)$. By (4.2), the Jacobian matrix of (4.1) at a positive equilibrium $(N_1,N_2)$ is

$$\begin{aligned} J = \left( \begin{array}{l@{\quad }c@{\quad }c@{\quad }r} - d_1N_1 &{} b_{12} \left( 1 + c s N_2^{s-1}\right) N_1 \hat{g}^2 \\ a_{21}N_2 \hat{g} &{}-a_{21} N_1N_2 \left( 1 + c s N_2^{s-1}\right) \hat{g}^2 \end{array} \right) , \end{aligned}$$

(6.4)

which implies tr$J(\hat{E}_{12}^\pm ) <0$. A direct computation shows that

$$\begin{aligned} \det J\left( \hat{E}_{12}^\pm \right) = \pm N_1N_2 \left( 1 + c s N_2^{s-1}\right) \hat{g}^2 \sqrt{ \Delta }. \end{aligned}$$

(6.5)

Thus, if $\Delta >0,$ then $\hat{E}_{12}^-$ is a saddle point and $\hat{E}_{12}^+$ is asymptotically stable.

When $\Delta =0,$ equilibria $\hat{E}_{12}^-$ and $\hat{E}_{12}^+$ coincide and $\det J(\hat{E}_{12}^-) =0$, which implies that there is a simple zero eigenvalue of $J(\hat{E}_{12}^-)$. We apply Sotomayor’s theorem to show that saddle-node bifurcation occurs at $\hat{E}_{12}^-$ when $\Delta =0,$ i.e., $a_{21} = a_{21}^0$. For the simple zero eigenvalue, we have the left and right eigenvectors of $J(\hat{E}_{12}^-)$ by (6.4):

$$\begin{aligned} w = \left( a_{21}N_2 \hat{g}, d_1N_1 \right) ^T,\quad v = \left( b_{12} \left( 1 + c s N_2^{s-1}\right) \hat{g}^2, d_1 \right) ^T. \end{aligned}$$

Let $\mu = a_{21}$ be the parameter in Sotomayor’s theorem. Let $\hat{F}=(\hat{F}_1,\hat{F}_2)^T$ be the righthand side of (4.1). Then $\hat{F}_\mu = (0, N_1N_2\hat{g} ) ^T,$ which implies that $w^T \hat{F}_\mu = d_1N_1^2N_2\hat{g}>0$ at $\hat{E}_{12}^-$ and $\mu =a_{21}^0$. Direct computations show that

$$\begin{aligned} \frac{\partial ^2 \hat{F}_1 }{\partial N_1^2}= & {} - 2 d_1,\quad \frac{\partial ^2 \hat{F}_2 }{\partial N_1^2} = 0,\quad \frac{\partial ^2 \hat{F}_1 }{\partial N_2^2} = b_{12} N_1 \hat{g}^2 \left[ c s \left( s-1\right) N_2^{s-2} -2 \hat{g} \left( 1 + c s N_2^{s-1}\right) ^2\right] , \\ \frac{\partial ^2 \hat{F}_1 }{\partial N_1 \partial N_2}= & {} b_{12} \left( 1 + c s N_2^{s-1}\right) \hat{g}^2,\quad \frac{\partial ^2 \hat{F}_2 }{\partial N_1 N_2} = a_{21} \hat{g} \left[ 1-N_2 \hat{g} \left( 1 + c s N_2^{s-1}\right) \right] , \\ \frac{\partial ^2 \hat{F}_2 }{\partial N_2^2}= & {} 2 r_2 \hat{g} \left( 1 + c s N_2^{s-1}\right) \left[ -1+ N_2 \hat{g} \left( 1 + c s N_2^{s-1}\right) \right] - r_2 \hat{g} c s \left( s-1\right) N_2^{s-2}. \end{aligned}$$

A long but straightforward computation shows that

$$\begin{aligned} w \cdot D^2\hat{F}(v,v) = - 2 d_1^2 a_{21} b_{12} N_1 N_2 \hat{g}^4 \left( 1 + c s N_2^{s-1}\right) <0. \end{aligned}$$

Thus, the Sotomayor’s theorem implies that saddle-node bifurcation occurs at $\hat{E}_{12}^-$ when $\Delta =0$. By a discussion similar to the proof for Theorem 3.3(iii), the result in (iii) is proven. $\square$