Article Contents
Article Contents

# Pattern formation of a predator-prey model with the cost of anti-predator behaviors

• * Corresponding author: Xingfu Zou.

Current address: Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, Canada

Research partially supported by the Natural Sciences and Engineering Research Council of Canada.
• We propose and analyse a reaction-diffusion-advection predator-prey model in which we assume that predators move randomly but prey avoid predation by perceiving a repulsion along predator density gradient. Based on recent experimental evidence that anti-predator behaviors alone lead to a 40% reduction on prey reproduction rate, we also incorporate the cost of anti-predator responses into the local reaction terms in the model. Sufficient and necessary conditions of spatial pattern formation are obtained for various functional responses between prey and predators. By mathematical and numerical analyses, we find that small prey sensitivity to predation risk may lead to pattern formation if the Holling type Ⅱ functional response or the Beddington-DeAngelis functional response is adopted while large cost of anti-predator behaviors homogenises the system by excluding pattern formation. However, the ratio-dependent functional response gives an opposite result where large predator-taxis may lead to pattern formation but small cost of anti-predator behaviors inhibits the emergence of spatial heterogeneous solutions.

Mathematics Subject Classification: 92D25, 34C23, 92D40.

 Citation:

• Figure 1.  Conditions of global stability of $E(\bar{u}, \bar{v})$ when $\alpha$ varies with $m(v) = m_1+m_2\, v$ and $p(u, v) = p.$ Parameters are: $r_0 = 5, \, a = 1, \, d = 0.2, \, p = 0.5, \, c = 0.5, \, m_1 = 0.3\, m_2 = 1, \, M = 10, d_u = 1, \, d_v = 2, \, k_0 = 1.$

Figure 2.  Conditions of global stability of $E(\bar{u}, \bar{v})$ when $k_0$ varies with $m(v) = m_1+m_2\, v$ and $p(u, v) = p.$ Parameters are: $r_0 = 5, \, a = 1, \, d = 0.2, \, p = 0.5, \, c = 0.5, \, m_1 = 0.3\, m_2 = 1, \, M = 10, d_u = 1, \, d_v = 2, \, \alpha = 0.5.$

Figure 3.  Spatial homogeneous steady states of $u, v$ with the Holling type Ⅱ functional response and density-dependent death function of predators when $\alpha$ is large. Parameters are: $r_0 =.8696, \, d =.1827, \, a =.6338, \, p =6.395, \, q =4.333, \, m_1 =0.72e-2, \, m_2 =.9816, \, c =.2645, \, d_u =0.2119e-1, \, d_v =1.531, \, \alpha =12, \, k_0 =0.1e-1, \, M =10, \, L = 4.$

Figure 4.  Spatial heterogeneous steady states of $u, v$ with the Holling type Ⅱ functional response and density-dependent death function of predators when $\alpha$ is small. Parameters are: $r_0 =.8696, \, d =.1827, \, a =.6338, \, p =6.395, \, q =4.333, \, m_1 =0.72e-2, \, m_2 =.9816, \, c =.2645, \, d_u =0.2119e-1, \, d_v =1.531, \, \alpha =8, \, k_0 =0.1e-1, \, M =10, \, L = 4.$

Figure 5.  Spatial homogeneous steady states of $u, v$ when $k_0 = 0,$ $\alpha$ is large, $m(v) = m_1,$ and $p(u, v) = b_1/(b_2\, v+u).$ Parameters are: $r_0 = 6.1885, \, d = 4.0730, \, a = 0.8481, \, b_1 = 4.5677, \, b_2 = 1.4380, \, m_1 = 1.6615, \, c = 0.9130, \, \alpha = 12, \, d_u = 0.0113, \, d_v = 4.7804, \, M = 10, \, k_0 = 0, \, L = 5.0212.$

Figure 6.  Spatial heterogenous steady states of $u, v$ when $k_0 = 0,$ $\alpha$ is small, $m(v) = m_1,$ and $p(u, v) = b_1/(b_2\, v+u).$ Parameters are: $r_0 = 6.1885, \, d = 4.0730, \, a = 0.8481, \, b_1 = 4.5677, \, b_2 = 1.4380, \, m_1 = 1.6615, \, c = 0.9130, \, \alpha = 5.1571, \, d_u = 0.0113, \, d_v = 4.7804, \, M = 10, \, k_0 = 0, \, L = 5.0212.$

Figure 7.  Conditions of diffusion-taxis-driven instability of $E(\bar{u}, \bar{v})$ with changing $\alpha$ when $k_0 \neq 0,$ $m(v) = m_1,$ and $p(u, v) = b_1/(b_2\, v+u).$ Parameters are: $r_0 =1.7939, \, d =0.2842, \, a =0.4373, \, b_1 =2.9354, \, b_2 =3.2998, \, m_1 =0.5614, \, c =0.6010, \, d_u =0.0344, \, d_v =7.2808, \, k_0 =8.0318, \, M = 10.$

Figure 8.  Spatial homogeneous steady states of $u,v$ when $k_0 \neq 0,$ $\alpha$ is small, $m(v)=m_1,$ and $p(u,v)=b_1/(b_2\,v+u).$ Parameters are: $r_0 = 1.7939,\, d = 0.2842,\, a = 0.4373,\, b_1 = 2.9354,\, b_2 = 3.2998,\, m_1 = 0.5614,\, c = 0.6010,\, d_u = 0.0344,\, d_v = 7.2808,\, k_0 = 8.0318,\, M=10,\,\alpha=0.3,\,L=2.6602.$

Figure 9.  Spatial heterogenous steady states of $u, v$ when $k_0 \neq 0,$ $\alpha$ is large, $m(v) = m_1,$ and $p(u, v) = b_1/(b_2\, v+u).$ Parameters are: $r_0 =1.7939, \, d =0.2842, \, a =0.4373, \, b_1 =2.9354, \, b_2 =3.2998, \, m_1 =0.5614, \, c =0.6010, \, d_u =0.0344, \, d_v =7.2808, \, k_0 =8.0318, \, M = 10, \, \alpha = 0.7957, \, L = 2.6602.$

Figure 10.  Spatial homogeneous steady states of $u, v$ when $k_0$ is small, $\alpha \neq 0,$ $m(v) = m_1,$ and $p(u, v) = b_1/(b_2\, v+u).$ Parameters are: $r_0 =1.7939, \, d =0.2842, \, a =0.4373, \, b_1 =2.9354, \, b_2 =3.2998, \, m_1 =0.5614, \, c =0.6010, \, d_u =0.0344, \, d_v =7.2808, \, k_0 =2, \, M = 10, \, \alpha = 0.7957, \, L = 2.6602.$

Figure 11.  Spatial homogeneous but temporal periodic solution $u, v$ over time when $m(v) = m_1+m_2\, v,$ $k_0$ is large, and $p(u, v) = b_1/(b_2\, v+u).$ Parameters are: $r_0 =4.8712, \, d =.9235, \, a =.9508, \, b_1 =.3433, \, b_2 =.6731, \, m_1 =0.228e-1, \, m_2 =.7908, \, c =.2959, \, d_u =.1516, \, d_v =8.5545, \, k_0 = 10, \, \alpha =7.4798, \, M = 10, \, L = 10.$

Figure 12.  Spatial homogeneous steady states of $u, v$ when $m(v) = m_1, k_0 \neq 0,$ $\alpha$ is large, and $p(u, v) = p/(1+q_1\, u+q_2\, v).$ Parameters are: $r_0 =.3558, \, d =0.832e-1, \, a =0.106e-1, \, p =.6313, \, q_1 =.4418,$ $q_2 =.3188, \, m_1 =.4901, \, c =.4780, \, d_u =0.324e-1, \, d_v =3.7446, \, M =100, \, \alpha = 0.1, \, k_0 = 1, L = 7$ .

Figure 13.  Spatial heterogeneous steady states of $u, v$ when $m(v) = m_1, k_0 \neq 0,$ $\alpha$ is small, and $p(u, v) = p/(1+q_1\, u+q_2\, v).$ Parameters are: $r_0 =.3558, \, d =0.832e-1, \, a =0.106e-1, \, p =.6313,$ $q_1 =.4418, \, q_2 =.3188, \, m_1 =.4901, \, c =.4780, \, d_u =0.324e-1, \, d_v =3.7446, \, M =100, \, \alpha = 0.01, \, k_0 = 1, L = 7$ .

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