Dynamics of Intraguild Predation Systems with Intraspecific Competition

Abstract

This paper considers intraguild predation (IGP) systems where species in the same community kill and eat each other and there is intraspecific competition in each species. The IGP systems are characterized by a lattice gas model, in which reaction between sites on the lattice occurs in a random and independent way. Global dynamics of the model with two species demonstrate mechanisms by which IGP leads to survival/extinction of species. It is shown that an intermediary level of predation promotes survival of species, while over-predation or under-predation could result in species extinction. An interesting result is that increasing intraspecific competition of one species can lead to extinction of one or both species, while increasing intraspecific competitions of both species would result in coexistence of species in facultative predation. Initial population densities of the species are also shown to play a role in persistence of the system. Then the analysis is extended to IGP systems with one species. Numerical simulations confirm and extend our results.

Appendix

Appendix A: Proof of Lemma 3.1

Let $f_1$ and $f_2$ be the right-hand sides of (1), respectively. Denote $B({X_1},{X_2}) = 1/({X_1}{X_2})$. Then we obtain

$$\begin{aligned} \frac{{\partial (B{f_1})}}{{\partial {X_1}}} + \frac{{\partial (B{f_2})}}{{\partial {X_2}}} = - \frac{ {{\epsilon _1} + {\sigma _1}{k_2}{X_2} + {c_1}} }{{{X_2}}} - \frac{ {{\epsilon _2} + {\sigma _2}{k_1}{X_1} + {c_2}} }{{{X_1}}} < 0. \end{aligned}$$

By Bendixson–Dulac Theorem (Perko 2001), there is no periodic orbit of (1).

If ${X_1} + {X_2} \ge 1$, then $d{X_1}/dt < 0$ and $d{X_2}/dt < 0$ by (1). Thus, the set $\{ ({X_1},{X_2}):{X_1} + {X_2} \le 1\}$ attracts all solutions of (1), which implies that solutions of (1) are bounded.

Appendix B: Proof of Lemma 3.2

Let $P({X_1},{X_2})$ be the intersection point of $l_1$ and $l_2$. From (5), a direct computation shows that $P(X_1,X_2)$ satisfies

$$\begin{aligned} {p_0}X_2^3 + {p_1}X_2^2 + {p_2}{X_2} + {p_3} = 0, \end{aligned}$$

(20)

where the expressions of $p_i$ are complex and are omitted here. Since the polynomial in (20) is cubic, the hyperbolas $l_1$ and $l_2$ have at most three intersection points on the plane.

Assume asymptotes $l_{11}$ and $l_{21}$ coincide, i.e., $a_1=a_2$. From (6) and (8), $X_1$ is a liner function of $X_2$. By (6), $X_1$ satisfies a quadratic equation, which implies that system (4) has at most two positive equilibria.

Assume asymptotes $l_{11}$ and $l_{21}$ do not coincide. Without loss of generality, let $a_1<a_2$. Then, $l_{11}$ is below $l_{21}$. When $e_1>0$, $e_2>0$, $l_1$ and $l_2$ have an intersection point in the fourth quadrant. When $e_1\le 0$, $e_2>0$, $l_1$ and $l_2$ have an intersection point in the third quadrant. When $e_1 > 0$, $e_2\le 0$, $l_1$ and $l_2$ have an intersection point in the fourth quadrant. When $e_1\le 0$, $e_2\le 0$, $l_1$ and $l_2$ have an intersection point in the third quadrant. Therefore, system (4) has at most two positive equilibria.

Appendix C: Proof of Lemma 3.3

We prove the result for $i=1$, while a similar one can be given for $i=2$. Since $a_1>0$, $l_1$ is convex rightward in the region ${X_2} > - {{{h_1}} / {{\sigma _1}}}$ and has a vertex ${\bar{P}}({{\bar{X}}_1},{{\bar{X}}_2})$ with

$$\begin{aligned} {{\bar{X}}_1} = {X_1}({{\bar{X}}_2},{\sigma _1}),~{{\bar{X}}_2} = \frac{{ - {c_1} - 1 + \sqrt{{c_1} - {\rho _1} + {\sigma _1}{c_1} + {\sigma _1}{m_1} - {\rho _1}{c_1} + {c_1}^2} }}{{{\sigma _1}}}. \end{aligned}$$

From $1-m_1\le 0$, we have ${l_1} \cap {\mathrm{int}} R_ + ^2 \ne \emptyset$ if and only if $l_1$ and the positive $X_2$-axis have two intersection points. That is, the following equation has two positive roots

$$\begin{aligned} {\tilde{a}}X_2^2 + {\tilde{b}}{X_2} + {\tilde{c}} = 0 \end{aligned}$$

where ${\tilde{a}} = {\sigma _1}$, ${\tilde{b}} = 1 - {\sigma _1} + {\rho _1}$, ${\tilde{c}} = {m_1} - 1$. Thus we have ${\sigma _1} > \sigma _1^{(1)}$ where ${\sigma _1} = \sigma _1^{(1)}({\rho _1})$ is a monotonically increasing function of $\rho _1$ (i.e.,$k_1$).

Appendix D: Proof of Theorem 3.4

Since $1-m_1=1-m_2=0$, we obtain $\sigma _i^{(1)} = {\rho _i} + 1$ and $O \in {l_1} \cap {l_2}$. A direct computation shows that $1/s_1$ and $s_2$ are slopes of $l_1$ and $l_2$ at O, respectively. From ${\sigma _i} > \sigma _i^{(1)}$, we have $s_i>0$ and ${l_i} \cap {\mathrm{int}} R_ + ^2 \ne \emptyset ,~i = 1,2$. If $1/s_1<s_2$, it follows from the convexity of $l_1$ and $l_2$ that they have a unique intersection point $P^+$ in the first quadrant. By Lemma 3.1, $P^+$ is globally asymptotically stable. In other situations, $l_1$ and $l_2$ have no intersection point in the first quadrant. Thus, all positive solutions of (4) converge to O.

Appendix E: Proof of Theorem 3.5

If ${\sigma _1} < \sigma _1^{(2)}$, $l_1$ and $l_2$ interest at $P^-$ and $P^+$. Let $k_1^+$ (resp. $k_2^+$) denote the slope of $l_1$ (resp. $l_2$) at $P^+$. From the expression of $F_i$ in (5), we have

$$\begin{aligned} \frac{1}{{k_1^ + }} = - \frac{{\partial {F_1}}}{{\partial {X_2}}}/\frac{{\partial {F_1}}}{{\partial {X_1}}}{|_{{P^ + }}},~k_2^ + = - \frac{{\partial {F_2}}}{{\partial {X_1}}}/\frac{{\partial {F_2}}}{{\partial {X_2}}}{|_{{P^ + }}} \end{aligned}$$

where

$$\begin{aligned} \frac{{\partial {F_1}}}{{\partial {X_1}}}{|_{{P^ + }}} = - (1 + {\sigma _1}{X_2} + {c_1})< 0,~\frac{{\partial {F_2}}}{{\partial {X_2}}}{|_{{P^ + }}} = - (1 + {\sigma _2}{X_1} + {c_2}) < 0. \end{aligned}$$

Stability of $P^+$ is considered as follows. The Jacobian matrix of (4) at $P^+$ is

$$\begin{aligned} J({P^ + }) = {\left( {\begin{array}{*{20}{c}} {{\epsilon _1}{X_1}\frac{{\partial {F_1}}}{{\partial {X_1}}}}&{}{{\epsilon _1}{X_1}\frac{{\partial {F_1}}}{{\partial {X_2}}}}\\ {{\epsilon _2}{X_2}\frac{{\partial {F_2}}}{{\partial {X_1}}}}&{}{{\epsilon _2}{X_2}\frac{{\partial {F_2}}}{{\partial {X_2}}}} \end{array}} \right) _{{P^ + }}}. \end{aligned}$$

Then

$$\begin{aligned} trJ({P^ + }) = \left( {{\epsilon _1}{X_1}\frac{{\partial {F_1}}}{{\partial {X_1}}} + {\epsilon _2}{X_2}\frac{{\partial {F_2}}}{{\partial {X_2}}}} \right) {|_{{P^ + }}} < 0. \end{aligned}$$

Moreover, we have

$$\begin{aligned} \det J({P^ + })= & {} {\epsilon _1}{\epsilon _2}{X_1}{X_2}\left( \frac{{\partial {F_1}}}{{\partial {X_1}}}\frac{{\partial {F_2}}}{{\partial {X_2}}} - \frac{{\partial {F_1}}}{{\partial {X_2}}}\frac{{\partial {F_2}}}{{\partial {X_1}}}\right) \\= & {} {\epsilon _1}{\epsilon _2}{X_1}{X_2}\frac{{\partial {F_1}}}{{\partial {X_1}}}\frac{{\partial {F_2}}}{{\partial {X_2}}}{|_{{p^ + }}}\left( 1 - \frac{{k_2^\mathrm{{ + }}}}{{k_1^ + }}\right) . \end{aligned}$$

Assume $k_1^+<0$. Then, $P^+$ is above the vertex of $l_1$. Notice that $l_1$ and the positive $X_2$-axis form a convex area ${\Omega _1}$. At the point $P^+$, $l_2$ interests with $l_1$ from the inside of ${\Omega _1}$ to its outside. Thus, if $k_2^+<0$, then we have $k_1^ + < k_2^ +$, which implies that $\det J\left( {{P^ + }} \right) > 0$. If $k_2^ + > 0$, we can see that $\det J\left( {{P^ + }} \right) > 0$.

Assume $k_1^ + > 0$. Then, $P^+$ is below the vertex of $l_1$. When $k_2^ + > 0$, we have $k_1^ + > k_2^ +$, which implies that $\det J\left( {{P^ + }} \right) > 0$. When $k_2^ + \le 0$, we can see that $\det J\left( {{P^ + }} \right) > 0$. It follows from convexity of $l_1$ that $k_1^ + \ne 0$. Thus, we have $\det J\left( {{P^ + }} \right) > 0$, which implies that $P^+$ is asymptotically stable.

Similarly, let $k_1^-$ (resp. $k_2^-$) denote the slope of $l_1$ (resp. $l_2$) at $P^-$. Then, we have $k_1^ - < k_2^ -$ and $\det J\left( {{P^ - }} \right) < 0$, which implies that $P^-$ is a saddle point. Thus the stable manifold of $P^-$ subdivide the first quadrant into two regions, one is the basin of attraction of O while the other is that of $P^+$.

When ${\sigma _2} = \sigma _2^{(2)}$, $P^-$ and $P^+$ coincide and form a saddle-node point (Perko 2001). Thus, separatrices of the saddle point subdivide the first quadrant into two regions, one is the basin of attraction of O while the other is that of $P^+$. A similar discussion can be given for $l_1$ and $\sigma _1^{(2)}$.

If ${\sigma _1} \le \sigma _1^{(1)}$ or ${\sigma _2} < \sigma _2^{(2)}$, there is no positive equilibrium of (4), which implies that all positive solutions of (4) converge to O.

Appendix F: Proof of Theorem 3.6

From $m_2=1$, we have $O \in {l_2}$. If $\sigma _2>\sigma _2^{(3)}$, then $P_1$ is below $l_2$ and is a saddle point. It follows from the convexity of $l_1$ and $l_2$ that they have a unique intersection point $P^+$ in the first quadrant. Thus system (4) has a unique positive equilibrium $P^+$. By a proof similar to that of Theorem 3.5, $P^+$ is globally asymptotically stable. If $\sigma _2\le \sigma _2^{(3)}$, system (4) has no positive equilibrium, which implies that all positive solutions of (4) converge to $P_1$.

Appendix G: Proof of Theorem 3.10

From ${\rho _1} < {{{\sigma _1}({m_1} + {c_1})} / {(1 + {c_1})}} + {c_1}$, we have ${\sigma _1} > \sigma _1^{(3)}$. Thus $P_2$ is at the left of $l_1$ and is a saddle point. If ${\sigma _2} > \sigma _2^{(3)}$, $P_1$ is below $l_2$ and is a saddle point. Thus, $l_1$ and $l_2$ have intersection points in the first quadrant. Moreover, since $l_1$ and $l_2$ have asymptotes ${X_2} = - {{(1 + {c_1})} / {{\sigma _1}}}$, ${X_1} = - {{(1 + {c_2})} / {{\sigma _2}}}$, respectively, they have an intersection point in the third quadrant. Thus, $l_1$ and $l_2$ have a unique intersection point in the first quadrant by Lemma 3.2, and system (4) has a unique positive equilibrium $P^+$. By a proof similar to that of Theorem 3.5, $P^+$ is globally asymptotically stable. In other situations, system (4) has no positive equilibrium. Thus, $P_1$ is globally asymptotically stable.

Appendix H: Proof of Theorem 3.14

From ${\rho _2} \ge {{{\sigma _2}({m_2} + {c_2})} / {(1 + {c_2})}} + {c_2}$, we have $e_2\le 0$ and ${\sigma _2} < \sigma _2^{(3)}$, which implies that $l_2$ is convex downward and $P_1$ is above $l_2$. If ${\sigma _1} \ge \sigma _1^{(3)}$, $P_2$ is at the left of $l_1$. By the convexity of $l_1$ and $l_2$ and Lemma 3.13, system (4) has no positive equilibrium. By a proof similar to that of Theorem 3.5, $P_1$ is globally asymptotically stable.

If ${\sigma _1} < \sigma _1^{(3)}$, then $P_2$ is at the right of $l_1$. Thus system (4) has a unique positive equilibrium $P^-$, which is a saddle point. The separatrices of the saddle point subdivide the first quadrant into two regions, one is the basin of attraction of $P_1$ while the other is that of $P_2$.