# American Institute of Mathematical Sciences

• Previous Article
A multi-base harmonic balance method applied to Hodgkin-Huxley model
• MBE Home
• This Issue
• Next Article
Analyzing the causes of alpine meadow degradation and the efficiency of restoration strategies through a mathematical modelling exercise
June  2018, 15(3): 775-805. doi: 10.3934/mbe.2018035

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

 Department of Applied Mathematics, University of Western Ontario, London, ON, Canada N6A 5B7

* Corresponding author: Xingfu Zou.

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

Received  September 21, 2016 Revised  June 07, 2017 Published  December 2017

Fund Project: 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.

Citation: Xiaoying Wang, Xingfu Zou. Pattern formation of a predator-prey model with the cost of anti-predator behaviors. Mathematical Biosciences & Engineering, 2018, 15 (3) : 775-805. doi: 10.3934/mbe.2018035
##### References:

show all references

##### References:
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: $r_0 = 5, \, a = 1, \, d = 0.2, \, p = 0.5, \, c = 0.5, \, m_1 = 0.3\, m_2 = 1, \, M = 10, d_u = 1, \, d_v = 2, \, k_0 = 1.$
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: $r_0 = 5, \, a = 1, \, d = 0.2, \, p = 0.5, \, c = 0.5, \, m_1 = 0.3\, m_2 = 1, \, M = 10, d_u = 1, \, d_v = 2, \, \alpha = 0.5.$
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: $r_0 =.8696, \, d =.1827, \, a =.6338, \, p =6.395, \, q =4.333, \, m_1 =0.72e-2, \, m_2 =.9816, \, c =.2645, \, d_u =0.2119e-1, \, d_v =1.531, \, \alpha =12, \, k_0 =0.1e-1, \, M =10, \, L = 4.$
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: $r_0 =.8696, \, d =.1827, \, a =.6338, \, p =6.395, \, q =4.333, \, m_1 =0.72e-2, \, m_2 =.9816, \, c =.2645, \, d_u =0.2119e-1, \, d_v =1.531, \, \alpha =8, \, k_0 =0.1e-1, \, M =10, \, L = 4.$
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: $r_0 = 6.1885, \, d = 4.0730, \, a = 0.8481, \, b_1 = 4.5677, \, b_2 = 1.4380, \, m_1 = 1.6615, \, c = 0.9130, \, \alpha = 12, \, d_u = 0.0113, \, d_v = 4.7804, \, M = 10, \, k_0 = 0, \, L = 5.0212.$
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: $r_0 = 6.1885, \, d = 4.0730, \, a = 0.8481, \, b_1 = 4.5677, \, b_2 = 1.4380, \, m_1 = 1.6615, \, c = 0.9130, \, \alpha = 5.1571, \, d_u = 0.0113, \, d_v = 4.7804, \, M = 10, \, k_0 = 0, \, L = 5.0212.$
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: $r_0 =1.7939, \, d =0.2842, \, a =0.4373, \, b_1 =2.9354, \, b_2 =3.2998, \, m_1 =0.5614, \, c =0.6010, \, d_u =0.0344, \, d_v =7.2808, \, k_0 =8.0318, \, M = 10.$
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: $r_0 = 1.7939,\, d = 0.2842,\, a = 0.4373,\, b_1 = 2.9354,\, b_2 = 3.2998,\, m_1 = 0.5614,\, c = 0.6010,\, d_u = 0.0344,\, d_v = 7.2808,\, k_0 = 8.0318,\, M=10,\,\alpha=0.3,\,L=2.6602.$
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: $r_0 =1.7939, \, d =0.2842, \, a =0.4373, \, b_1 =2.9354, \, b_2 =3.2998, \, m_1 =0.5614, \, c =0.6010, \, d_u =0.0344, \, d_v =7.2808, \, k_0 =8.0318, \, M = 10, \, \alpha = 0.7957, \, L = 2.6602.$
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: $r_0 =1.7939, \, d =0.2842, \, a =0.4373, \, b_1 =2.9354, \, b_2 =3.2998, \, m_1 =0.5614, \, c =0.6010, \, d_u =0.0344, \, d_v =7.2808, \, k_0 =2, \, M = 10, \, \alpha = 0.7957, \, L = 2.6602.$
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: $r_0 =4.8712, \, d =.9235, \, a =.9508, \, b_1 =.3433, \, b_2 =.6731, \, m_1 =0.228e-1, \, m_2 =.7908, \, c =.2959, \, d_u =.1516, \, d_v =8.5545, \, k_0 = 10, \, \alpha =7.4798, \, M = 10, \, L = 10.$
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: $r_0 =.3558, \, d =0.832e-1, \, a =0.106e-1, \, p =.6313, \, q_1 =.4418,$ $q_2 =.3188, \, m_1 =.4901, \, c =.4780, \, d_u =0.324e-1, \, d_v =3.7446, \, M =100, \, \alpha = 0.1, \, k_0 = 1, L = 7$ .
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: $r_0 =.3558, \, d =0.832e-1, \, a =0.106e-1, \, p =.6313,$ $q_1 =.4418, \, q_2 =.3188, \, m_1 =.4901, \, c =.4780, \, d_u =0.324e-1, \, d_v =3.7446, \, M =100, \, \alpha = 0.01, \, k_0 = 1, L = 7$ .
 [1] Yinshu Wu, Wenzhang Huang. Global stability of the predator-prey model with a sigmoid functional response. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1159-1167. doi: 10.3934/dcdsb.2019214 [2] Wei Feng, Jody Hinson. Stability and pattern in two-patch predator-prey population dynamics. Conference Publications, 2005, 2005 (Special) : 268-279. doi: 10.3934/proc.2005.2005.268 [3] Ming Liu, Dongpo Hu, Fanwei Meng. Stability and bifurcation analysis in a delay-induced predator-prey model with Michaelis-Menten type predator harvesting. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020259 [4] Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002 [5] Wonlyul Ko, Inkyung Ahn. Pattern formation of a diffusive eco-epidemiological model with predator-prey interaction. Communications on Pure & Applied Analysis, 2018, 17 (2) : 375-389. doi: 10.3934/cpaa.2018021 [6] Zhong Li, Maoan Han, Fengde Chen. Global stability of a predator-prey system with stage structure and mutual interference. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 173-187. doi: 10.3934/dcdsb.2014.19.173 [7] Jinfeng Wang, Sainan Wu, Junping Shi. Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1273-1289. doi: 10.3934/dcdsb.2020162 [8] Prabir Panja, Soovoojeet Jana, Shyamal kumar Mondal. Dynamics of a stage structure prey-predator model with ratio-dependent functional response and anti-predator behavior of adult prey. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 391-405. doi: 10.3934/naco.2020033 [9] Kexin Wang. Influence of feedback controls on the global stability of a stochastic predator-prey model with Holling type Ⅱ response and infinite delays. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1699-1714. doi: 10.3934/dcdsb.2019247 [10] Sílvia Cuadrado. Stability of equilibria of a predator-prey model of phenotype evolution. Mathematical Biosciences & Engineering, 2009, 6 (4) : 701-718. doi: 10.3934/mbe.2009.6.701 [11] Antoni Leon Dawidowicz, Anna Poskrobko. Stability problem for the age-dependent predator-prey model. Evolution Equations & Control Theory, 2018, 7 (1) : 79-93. doi: 10.3934/eect.2018005 [12] Evan C. Haskell, Jonathan Bell. Pattern formation in a predator-mediated coexistence model with prey-taxis. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2895-2921. doi: 10.3934/dcdsb.2020045 [13] Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979 [14] Dan Li. Global stability in a multi-dimensional predator-prey system with prey-taxis. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1681-1705. doi: 10.3934/dcds.2020337 [15] Mary Ballyk, Ross Staffeldt, Ibrahim Jawarneh. A nutrient-prey-predator model: Stability and bifurcations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 2975-3004. doi: 10.3934/dcdss.2020192 [16] Leonid Braverman, Elena Braverman. Stability analysis and bifurcations in a diffusive predator-prey system. Conference Publications, 2009, 2009 (Special) : 92-100. doi: 10.3934/proc.2009.2009.92 [17] Wei Feng, Michael T. Cowen, Xin Lu. Coexistence and asymptotic stability in stage-structured predator-prey models. Mathematical Biosciences & Engineering, 2014, 11 (4) : 823-839. doi: 10.3934/mbe.2014.11.823 [18] Yiwen Tao, Jingli Ren. The stability and bifurcation of homogeneous diffusive predator–prey systems with spatio–temporal delays. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021038 [19] Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041 [20] Jicai Huang, Yijun Gong, Shigui Ruan. Bifurcation analysis in a predator-prey model with constant-yield predator harvesting. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2101-2121. doi: 10.3934/dcdsb.2013.18.2101

2018 Impact Factor: 1.313