Article Contents
Article Contents

# Studying the fear effect in a predator-prey system with apparent competition

• * Corresponding author: Xiaoying Wang

The first author is supported by the NSERC of Canada (RGPIN-2020-06825 and DGECR-2020-00369)

• Recent experimental evidence shows that the mere presence of predators may largely reduce the reproduction success of prey. The loss of prey's reproduction rate is attributed to the cost of anti-predator defense of prey when the prey perceives predation risks. We propose a predator-prey model where the prey shares a common enemy that leads to apparent competition between the prey and also the cost of anti-predator defense. Analytical results give the persistence conditions for the population densities of the prey and the predator. Numerical simulations demonstrate rich dynamics, such as the bi-stability of an equilibrium and a limit cycle. Results also reveal how the prey and the predator may coexist when the anti-predator defense level varies in prey. A relatively strong anti-predator defense in the prey may drive the population density of the prey to extinction and change the original coexistence of all the prey and the predator where the population densities oscillate periodically. Alternatively, strong anti-predator defense in the prey may facilitate the coexistence of the prey and the predator at a steady state.

Mathematics Subject Classification: 34C23, 92D25.

 Citation:

• Figure 1.  Steady-state solutions for (1) with the linear functional response. Parameters are: 1(a) $r_1 = 2, k_1 = 1, d_1 = 0.2, a_1 = 0.1, p_1 = 0.4, r_2 = 5, $$k_2 = 2, d_2 = 0.3, a_2 = 0.2, p_2 = 0.3, c_1 = 0.6, c_2 = 0.3, m = 6.5; 1(b) r_1 = 2, k_1 = 1, d_1 = 0.2, a_1 = 0.1, p_1 = 0.4, r_2 = 5, k_2 = 10,$$ d_2 = 0.3, a_2 = 0.2, p_2 = 0.3, c_1 = 0.6, c_2 = 0.3, m = 1;$ 1(c) $r_1 = 2, k_1 = 1, d_1 = 0.2, a_1 = 0.1, p_1 = 0.4, r_2 = 5, k_2 = 2, $$d_2 = 0.3, a_2 = 0.2, p_2 = 0.3, c_1 = 0.6, c_2 = 0.3, m = 0.2; 1(d) r_1 = 2, k_1 = 1, d_1 = 0.2, a_1 = 0.1, p_1 = 0.4, r_2 = 5, k_2 = 2, d_2 = 0.3,$$a_2 = 0.2, p_2 = 0.3, c_1 = 0.6, c_2 = 0.3, m = 2.$ Initial condition for 1(a)–1(d) is $(u_1(0),u_2(0),v(0)) = (5,1,3).$

Figure 2.  Existence of the positive equilibrium $E(\hat{u}_1,\hat{u}_2,\hat{v})$ of (1) with the linear functional response under varying $k_1$ and $k_2.$ Parameters are: $r_1 = 8, d_1 = 0.2, a_1 = 1, p_1 = 0.6, r_2 = 6, d_2 = 1, a_2 = 0.2, p_2 = 0.2, c_1 = 0.6, c_2 = 0.1, m = 0.6.$

Figure 3.  Steady-state solutions for (1) with the linear functional response, when $k_1$ or $k_2$ varies. Parameters are: 3(a) $r_1 = 2, d_1 = 0.2, a_1 = 0.1, p_1 = 0.4, r_2 = 5, d_2 = 0.3,$$a_2 = 0.2, p_2 = 0.3, c_1 = 0.6, c_2 = 0.3, m = 2, k_2 = 1; 3(b) r_1 = 2, d_1 = 0.2,$$ a_1 = 0.1, p_1 = 0.4, r_2 = 5, d_2 = 0.3, a_2 = 0.2, p_2 = 0.3, c_1 = 0.6, c_2 = 0.3, m = 2, k_1 = 1.$

Figure 4.  Bi-stability of the boundary equilibrium $E_4$ and a limit cycle in the interior of the $u_1-u_2-v$ space. Parameters are: $r_1 = 6.92, k_1 = 5.5, d_1 = 0.24, a_1 = 0.22, p_1 = 0.38, h_1 = 3.8, r_2 = 5.15, k_2 = 9.11, d_2 = 0.09,$$a_2 = 0.06, p_2 = 0.36, h_2 = 2.45, c_1 = 0.44, c_2 = 0.4, m = 0.04. Initial condition for 4(a) is IC1 = (2.6, 0.5, 1.3). Initial condition for 4(b) is IC2 = (0, 0.8, 1.6). Initial condition for 4(c) is IC3 = (5, 5, 5). Figure 5. Basin of attraction for the boundary equilibrium E_4 and the positive periodic solution. Parameters are the same as Figure 4 except the initial conditions IC = (u_1(0),u_2(0),v(0)). In 5(a), u_2(0) = 0.5; in 5(b), v(0) = 2. Figure 6. Oscillatory solutions of (1) with the Holling type Ⅱ functional response when k_1 varies. Parameters are identical to Figure 4 except k_1. For 6(a), k_1 = 7.5 and for 6(b), k_1 = 10. The initial condition is IC = (5,5,5). Figure 7. Bi-stability of the boundary equilibrium E_5 and a periodic solution on the boundary. Parameters are: r_1 = 6.2, d_1 = 0.63, a_1 = 0.11, p_1 = 0.35, h_1 = 1.48, r_2 = 5.85,$$d_2 = 0.25, a_2 = 0.23, p_2 = 0.66, h_2 = 1.34, c_1 = 0.98, c_2 = 0.29,$$m = 0.12, k_1 = 1, k_2 = 0.8. Initial condition of 7(a) is IC1 = (0.2,3.9,3.5). Initial condition of 7(b) is IC2 = (0.5,3.9,3.5). Figure 8. Periodic solutions/steady-state solutions of (1) with the Holling type Ⅱ functional response when k_2 varies. Parameters are: r_1 = 3.25, d_1 = 0.09, a_1 = 0.2, p_1 = 0.18, h_1 = 0.01,$$r_2 = 7, d_2 = 0.8, a_2 = 0.22, p_2 = 0.5, h_2 = 0.25, c_1 = 0.19, c_2 = 0.44, m = 0.3, k_1 = 1.$ For 8(a), $k_2 = 0.3$; for 8(b), $k_2 = 0.52$; for 8(c), $k_2 = 1.$ Initial condition is $IC = (5,5,5).$

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