# American Institute of Mathematical Sciences

• Previous Article
Asymmetric diffusion in a two-patch mutualism system characterizing exchange of resource for resource
• DCDS-B Home
• This Issue
• Next Article
Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback
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  February 2021 Early access  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
##### References:

show all references

##### References:
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
]. 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">Figure 8.  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
 [1] Jim M. Cushing. The evolutionary dynamics of a population model with a strong Allee effect. Mathematical Biosciences & Engineering, 2015, 12 (4) : 643-660. doi: 10.3934/mbe.2015.12.643 [2] Chuang Xu. Strong Allee effect in a stochastic logistic model with mate limitation and stochastic immigration. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2321-2336. doi: 10.3934/dcdsb.2016049 [3] Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051 [4] Dianmo Li, Zhen Zhang, Zufei Ma, Baoyu Xie, Rui Wang. Allee effect and a catastrophe model of population dynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 629-634. doi: 10.3934/dcdsb.2004.4.629 [5] Wenjie Ni, Mingxin Wang. Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3409-3420. doi: 10.3934/dcdsb.2017172 [6] Elena Braverman, Alexandra Rodkina. Stochastic difference equations with the Allee effect. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 5929-5949. doi: 10.3934/dcds.2016060 [7] Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129 [8] Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045 [9] Eduardo González-Olivares, Betsabé González-Yañez, Jaime Mena-Lorca, José D. Flores. Uniqueness of limit cycles and multiple attractors in a Gause-type predator-prey model with nonmonotonic functional response and Allee effect on prey. Mathematical Biosciences & Engineering, 2013, 10 (2) : 345-367. doi: 10.3934/mbe.2013.10.345 [10] J. Leonel Rocha, Abdel-Kaddous Taha, Danièle Fournier-Prunaret. Explosion birth and extinction: Double big bang bifurcations and Allee effect in Tsoularis-Wallace's growth models. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3131-3163. doi: 10.3934/dcdsb.2015.20.3131 [11] Na Min, Mingxin Wang. Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1721-1737. doi: 10.3934/dcdsb.2018073 [12] Yuying Liu, Yuxiao Guo, Junjie Wei. Dynamics in a diffusive predator-prey system with stage structure and strong allee effect. Communications on Pure & Applied Analysis, 2020, 19 (2) : 883-910. doi: 10.3934/cpaa.2020040 [13] Yujing Gao, Bingtuan Li. Dynamics of a ratio-dependent predator-prey system with a strong Allee effect. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2283-2313. doi: 10.3934/dcdsb.2013.18.2283 [14] Moitri Sen, Malay Banerjee, Yasuhiro Takeuchi. Influence of Allee effect in prey populations on the dynamics of two-prey-one-predator model. Mathematical Biosciences & Engineering, 2018, 15 (4) : 883-904. doi: 10.3934/mbe.2018040 [15] Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065 [16] Nika Lazaryan, Hassan Sedaghat. Extinction and the Allee effect in an age structured Ricker population model with inter-stage interaction. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 731-747. doi: 10.3934/dcdsb.2018040 [17] Nalin Fonseka, Jerome Goddard II, Ratnasingham Shivaji, Byungjae Son. A diffusive weak Allee effect model with U-shaped emigration and matrix hostility. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5509-5517. doi: 10.3934/dcdsb.2020356 [18] Yi Yang, Robert J. Sacker. Periodic unimodal Allee maps, the semigroup property and the $\lambda$-Ricker map with Allee effect. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 589-606. doi: 10.3934/dcdsb.2014.19.589 [19] Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256 [20] J. Leonel Rocha, Danièle Fournier-Prunaret, Abdel-Kaddous Taha. Strong and weak Allee effects and chaotic dynamics in Richards' growths. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2397-2425. doi: 10.3934/dcdsb.2013.18.2397

2020 Impact Factor: 1.327