Dynamics of an intraguild predation food web model with strong Allee effect in the basal prey

Abstract

Since intraguild predation (IGP) is a ubiquitous and important community module in nature and Allee effect has strong impact on population dynamics, in this paper we propose a three-species IGP food web model consisted of the IG predator, IG prey and basal prey, in which the basal prey follows a logistic growth with strong Allee effect. We investigate the local and global dynamics of the model with emphasis on the impact of strong Allee effect. First, positivity and boundedness of solutions are studied. Then existence and stability of the boundary and interior equilibria are presented and the Hopf bifurcation curve at an interior equilibrium is given. The existence of a Hopf bifurcation curve indicates that if competition between the IG prey and IG predator for the basal resource lies below the curve then the interior equilibrium remains stable, while if it lies above the curve then the interior equilibrium loses its stability. In order to explore the impact of Allee effect, the parameter space is classified into sixteen different regions and, in each region, the number of interior equilibria is determined and the corresponding bifurcation diagrams on the Allee threshold are given. The extinction parameter regions of at least one species and the necessary coexistence parameter regions of all three species are provided. In addition, we explore possible dynamical patterns, i.e., the existence of multiple attractors. By theoretical analysis and numerical simulations, we show that the model can have one (i.e. extinction of all species), two (i.e. bi-stability) or three (i.e. tri-stability) attractors. It is also found by simulations that when there exists a unique stable interior equilibrium, the model may generate multiple attracting periodic orbits and the coexistence of all three species is enhanced as the competition between the IG prey and IG predator for the basal resource is close to the Hopf bifurcation curve from below. Our results indicate that the intraguild predation food web model exhibits rich and complex dynamic behaviors and strong Allee effect in the basal prey increases the extinction risk of not only the basal prey but also the IG prey or/and IG predator.

Introduction

Intraguild predation (IGP) is defined as the killing and eating of potential competitors and is a combination of predation and competition [1]. A simple example of an IGP food web is the tri-trophic community module including a predator population (IG predator) and its prey (IG prey) sharing a common resource. Since the IG predator feeds on different trophic levels (the IG prey and their common basal resource), and simultaneously competes with another species (its IG prey), it is a specific case of omnivory [2], [3], [4], [5]. IGP is an important community module to understand the mechanism for persistence of complex food webs. Because of the ubiquity and importance of this interaction in nature, IGP has received considerable attention [6], [7], [8], [9], [10].

Various IGP models have been proposed and studied by many researchers. Holt and Polis [11] formed a three-species Lotka–Volterra type IGP model with Holling Type I functional response and showed that increase in the strength of intraguild predation could destabilize the positive equilibrium. Tanabe and Nambe [12] also considered an IGP model with the same functional response as in [11] and observed that intraguild predation might destabilize the system and induce chaos by numerical simulations. Hsu, Ruan, and Yang [2] considered a three-species food web model with Lotka–Volterra type interaction between populations, classified the parameter space into three categories containing eight cases, and demonstrated extinction results for five cases and verified uniform persistence for the other three cases. For more studies on the dynamics of IGP models, including ODE models, PDE models and delay models, we refer to [13], [14], [15], [16], [17], [18], [19], [20], [21] and the references therein.

In many studies of IGP models, see for example [2], [3], [11], [12], [14], [15], [16], [18], [19], the common prey of the IG predator and IG prey is assumed to follow the logistic growth. Although a logistic growth function can better depict individual population growth and has become extremely popular, but in real natural situation there are abundant evidences showing that, unlike the logistic growth, populations at low densities are influenced by positive relationship between the growth rate and the density of the population [22], [23], [24], [25], [26], [27]. This biological phenomenon is known as Allee effect [23], [25], [27], [28] and occurs when the species engages in social behavior such as cooperative hunting or group defense [22], [23], [29], [30], [31], [32].

A simple model with Allee effect takes the form

$$\frac{dX}{dt}=rX(K-X)(X-K_0),$$

where $r>0,K>0$ and $\left|K_0\right|<K$. The term $X-K_0$ is included as a modification of the logistic model. When $0<K_0<K$, $K_0$ is a threshold population level (called Allee threshold), below which the population declines to extinction while above which the population persists. In this case, Eq. (1.1) describes the strong Allee effect [33], [34], [35], [36], [37]. If $K_0\le 0$, Eq. (1.1) represents the weak Allee effect [38]. A population with weak Allee effect does not have a critical threshold. Allee effect can result in the increase of the likelihood of extinction. Recently, Allee effect has attracted much attention owing to its strong potential impact on population dynamics and there are several different ways to model strong Allee effect (e.g. see [33], [34], [39], [40], [41], [42], [43], [44], [45], [46], [47]).

In this paper, we consider a three-species intraguild predation food web model which includes a predator population (IG predator) and its prey (IG prey) sharing a common prey. It is assumed that the shared prey exhibits strong Allee effect which is formulated by following [33], [34], [35], [36], [37]. The IGP food web model is represented as follows:

$$\begin{array}{c}\frac{dR}{dt}=rR\left(1-\frac{R}{K}\right)\left(R-K_0\right)-b_{1}RP-b_{2}RQ,\hfill \\ \frac{dP}{dt}=P(e_1{b}_{1}R-b_{3}Q-d_p),\hfill \\ \frac{dQ}{dt}=Q(e_2{b}_{2}R+e_3{b}_{3}P-d_q),\hfill \end{array}$$

where $R\left(t\right),P\left(t\right)$ and $Q\left(t\right)$ denote the densities of the shared prey, IG prey and IG predator at time $t$, respectively. All parameters are positive. $r$ and $K$ are the intrinsic growth rate and carrying capacity of the shared prey $R$, respectively; $K_0$ is the Allee threshold satisfying $0<K_0<K$; $d_p$ and $d_q$ are death rates of the IG prey $P$ and IG predator $Q$, respectively; $b_1$ and $b_2$ are predation rates of species $P$ and $Q$ to the shared prey $R$, respectively; $b_3$ is the predation rate of the IG predator $Q$ to IG prey $P$; $e_1$ and $e_2$ are conversion rates of resource consumption into reproduction for species $P$ and $Q$, respectively; $e_3$ is the conversion rate of the IG predator from the IG prey.

System (1.2) can be used to model many IGP food webs with strong Allee effect such as the predatory invertebrates–planktivorous fish–herbivorous zooplankton system, in which both predatory invertebrates and planktivorous fish feed on herbivorous zooplankton, while planktivorous fish also feeds on predatory invertebrates [48]. Sarnelle and Knapp [49] showed that the zooplankton suffers a strong Allee effect.

For mathematical simplification, we rewrite model (1.2) in a nondimensional form. Let $x=\frac{R}{K},y=P,z=Q$ and $\tau =rKt$. Then (1.2) takes the form

$$\begin{array}{c}{x}^{\prime }=x(x-\theta )(1-x)-\alpha xy-\beta xz,\hfill \\ y^{\prime }={\gamma }_{1}y(x-a_{1}z-d_1),\hfill \\ z^{\prime }={\gamma }_{2}z(x+a_{2}y-d_2),\hfill \end{array}$$

where $\theta =\frac{K_0}{K}$, $0<\theta <1$, and $\alpha =\frac{b_1}{rK},\beta =\frac{b_2}{rK},{\gamma }_i=\frac{e_i{b}_i}{r}(i=1,2),a_1=\frac{b_3}{e_1{b}_{1}K},a_2=\frac{e_3{b}_3}{e_2{b}_{2}K},d_1=\frac{d_p}{e_1{b}_{1}K},d_2=\frac{d_q}{e_2{b}_{2}K}$.

We will provide detailed mathematical analysis of model (1.3) with related biological implications. The main purpose of this article is to investigate the following two questions: First, how does Allee effect affect the dynamics of intraguild predation? Second, in the presence of Allee effect on the shared prey, under what conditions will the shared prey, IG prey and IG predator coexist?

To answer these two questions, we first show the positive invariance and boundedness of model (1.3) in Section 2. In order to understand the dynamics of (1.3), in Section 3 we first discuss the local and global properties of subsystems of (1.3). Then in Section 4 we investigate the existence and local stability of boundary equilibria and interior equilibria as well as the existence of Hopf bifurcation. The extinction of at least one species of the basal prey $x$, IG prey $y$ and IG predator $z$ is also studied in Section 4. Our results indicate that Allee effect in the basal prey increases the extinction risk of not only the basal prey but also the IG prey or/and IG predator. In Section 5 we explore the impact of Allee effect on the dynamics of model (1.3) in detail. The parameter space of $\lambda =(\alpha ,\beta ,a_1,a_2,d_1,d_2)$ is divided into sixteen different regions, and in each region the number of interior equilibria is determined and the corresponding bifurcation diagrams on the Allee threshold $\theta$ are given. The extinction parameter regions of at least one species and the necessary coexistence parameter regions of all three species are obtained. In Section 6, we focus on the possible dynamical patterns, i.e., the existence of multiple attractors, and their biological implications. It is shown that model (1.3) can have one (i.e. extinction of all species), two (i.e. bi-stability) or three (i.e. tri-stability) attractors. We also find by simulations that the orbits which tend to the extinction state and the stable interior equilibrium may be attracted to some periodic orbits as $({\gamma }_1,{\gamma }_2)$ is close to the Hopf bifurcation curve from below, and thus multiple attracting periodic orbits are generated and the coexistence of all three species is enhanced. In Section 7, we briefly make a comparison between the dynamics of model (1.3) and the dynamics of the IGP model without Allee effect in the basal prey, and provide a summary of our results.