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February  2021, 26(2): 943-962. doi: 10.3934/dcdsb.2020148

## A modified May–Holling–Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator

 1 School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, GP Campus, Brisbane, Queensland 4001 Australia, Facultad de Educación, Universidad de Las Américas, Av. Manuel Montt 948, Santiago, Chile 2 Department of Computer Science, The University of South Dakota, Vermillion, SD 57069, South Dakota, USA 3 School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, GP Campus, Brisbane, Queensland 4001 Australia

* Corresponding author: Claudio Arancibia-Ibarra

Received  May 2019 Revised  November 2019 Published  May 2020

We study a predator-prey model with Holling type Ⅰ functional response, an alternative food source for the predator, and multiple Allee effects on the prey. We show that the model has at most two equilibrium points in the first quadrant, one is always a saddle point while the other can be a repeller or an attractor. Moreover, there is always a stable equilibrium point that corresponds to the persistence of the predator population and the extinction of the prey population. Additionally, we show that when the parameters are varied the model displays a wide range of different bifurcations, such as saddle-node bifurcations, Hopf bifurcations, Bogadonov-Takens bifurcations and homoclinic bifurcations. We use numerical simulations to illustrate the impact changing the predation rate, or the non-fertile prey population, and the proportion of alternative food source have on the basins of attraction of the stable equilibrium point in the first quadrant (when it exists). In particular, we also show that the basin of attraction of the stable positive equilibrium point in the first quadrant is bigger when we reduce the depensation in the model.

Citation: Claudio Arancibia-Ibarra, José Flores, Michael Bode, Graeme Pettet, Peter van Heijster. A modified May–Holling–Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 943-962. doi: 10.3934/dcdsb.2020148
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In the left panel, we show the per capita growth rate of the logistic function (blue line), the strong Allee effect with $m = 0.1$ (red curve), the weak Allee effect with $m = -0.1$ (orange curve), multiple Allee effects with $m = 0.1$ and $b = 0.15$ (grey curve) and multiple Allee effects with $m = 0.1$ and $b = 0.05$ (green curve). In the right panel, we show the size of the depensation region for the strong Allee effect (6) (red curve) and for the multiple Allee effects (5) (grey curve) as function of the non-fertile prey population $b$. We observe that the depensation region for the multiple Allee effects is always smaller than the depensation region for the strong Allee effect
The intersections of the functions $p(u)$ (red line) and $d(u)$ (blue lines) for three different possible cases: (a) If $\Delta<0$ (10) then $p(u)$ and $d(u)$ do not intersect, and (8) does not have positive equilibrium points; (b) If $\Delta = 0$ then $p(u)$ and $d(u)$ intersect in one point, and (8) has a unique positive equilibrium point; (c) If $\Delta>0$ then $p(u)$ and $d(u)$ intersect in two points, and (8) has two distinct positive equilibrium points
Phase plane of system (8) and its invariant regions $\Phi$ and $\Gamma\backslash\Phi$
For $M = 0.05$, $B = 0.05$, $C = 0.5$, $Q = 0.8$, and $S = 0.175$, such that $\Delta<0$ (10), the equilibrium point $(0,C)$ is a global attractor for trajectories starting in the first quadrant. The blue (red) curve represents the prey (predator) nullcline
Let the system parameter $(M,B,C,Q) = (0.07,0.0645,0.32,0.736)$ be such that $\Delta>0$ (10). (a) If $S = 0.15$ such that $C<C_{H}$, then the equilibrium point $P_2$ is stable. (b) If $S = 0.05$ such that $C>C_{H}$, then the equilibrium point $P_2$ is unstable. The blue (red) curve represents the prey (predator) nullcline. The orange (light blue) region represents the basin of attraction of the equilibrium point $(0,C)$ ($P_2$). Note that the same color conventions are used in the upcoming figures
If $M = 0.05$, $B = 0.05$, $S = 0.125$ and $Q = 0.60821818$, then $\Delta = 0$. Therefore, the equilibrium point $P_3$ is (a) a saddle-node repeller if $C>C_{SN}$ and (b) a saddle-node attractor if $C<C_{SN}$
For $M = 0.05$, $B = 0.05$, $C = 0.58951256$, $S = 0.125$ and $Q = 0.60821818$, such that $\Delta = 0$ and $f(u_3) = C_{SN}$, the point $(0,C)$ is an attractor and the equilibrium point $P_3$ is a cusp point
The bifurcation diagram of system (8) for $M = 0.05$ and $S = 0.071080895$ fixed and created with the numerical bifurcation package MATCONT [17]. In the left panel $B = 0.1$ fixed and varying $Q$ and $C$ and in the right panel $Q = 0.608$ fixed and varying $B$ and $C$. The curve $C_H$ represents the Hopf curve, $C_{HOM}$ represents the homoclinic curve, $C_{SN}$ represents the saddle-node curve, and $BT$ represents the Bogdanov-Takens bifurcation.The corresponding phase planes for the different regions are shown in Figure 9
The phase planes of system (8) for $B = 0.1$, $M = 0.05$, $Q = 0.75$ and $S = 0.071080895$ fixed and varying $C$. This last parameter impacts the number of equilibrium points of system (8). The light blue area in the phase plane represent the basins of attraction of the equilibrium points $P_2$, while the orange area in the phase plane represent the basins of attraction of the equilibrium points $(0,C)$
The size of the basin of attraction of $p_2$, in units$^2$, of the stable equilibrium point $p_2$ of system (7) considering strong Allee effect (red line) and multiple Allee effect (blue line) for varying the non-fertile population $b$ and with other system parameters $r = 14$, $K = 150$, $m = 15$, $q = 1.08$, $s = 1.25$, $n = 0.05$ and $c = 0.75$ fixed. The blue dotted-dashed line represents the region where the stable manifold of the saddle equilibrium point $p_1$ connects with (K, 0) and the blue dashed line represent the region where the equilibrium point $p_2$ is surrounded by an unstable limit cycle
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