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January  2022, 27(1): 229-243. doi: 10.3934/dcdsb.2021038

The stability and bifurcation of homogeneous diffusive predator–prey systems with spatio–temporal delays

Yiwen Tao and  Jingli Ren ,

School of Mathematics and Statistics, Henan Academy of Big Data, Zhengzhou University, Zhengzhou 450001, China

* Corresponding author: renjl@zzu.edu.cn

Received  August 2020 Revised  December 2020 Published  January 2022 Early access  January 2021

In this paper, we consider a generalized predator-prey system described by a reaction-diffusion system with spatio-temporal delays. We study the local stability for the constant equilibria of predator-prey system with the generalized delay kernels. Moreover, using the specific delay kernels, we perform a qualitative analysis of the solutions, including existence, uniqueness, and boundedness of the solutions; global stability, and Hopf bifurcation of the nontrivial equilibria.

Keywords: Spatio-temporal delay, diffusion, stability, bifurcation, predator prey.
Mathematics Subject Classification: Primary: 35K57.
Citation: Yiwen Tao, Jingli Ren. The stability and bifurcation of homogeneous diffusive predator–prey systems with spatio–temporal delays. Discrete & Continuous Dynamical Systems - B, 2022, 27 (1) : 229-243. doi: 10.3934/dcdsb.2021038
References:
[1]

L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst. Ser. A, 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.  Google Scholar

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R. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differential Equations, 6 (2016), 3305-3343.  doi: 10.1016/j.jde.2016.05.025.  Google Scholar

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S. A. Gourley and M. V. Bartuccelli, Parameter domains for instability of uniform states in systems with many delays, J. Math. Biol., 35 (1997), 843-867.  doi: 10.1007/s002850050080.  Google Scholar

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S. A. Gourley and N. F. Britton, A predator-prey reaction-diffusion system with nonlocal effects, J. Math. Biol., 34 (1996) 297–333. doi: 10.1007/BF00160498.  Google Scholar

[8]

S. A. Gourley and S. Ruan, Convergence and travelling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. Anal., 35 (2003), 806-822.  doi: 10.1137/S003614100139991.  Google Scholar

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S. A. Gourley and J. W.-H. So, Dynamics of a food-limited population model incorporating nonlocal delays on a finite domain, J. Math. Biol., 44 (2002), 49-78.  doi: 10.1007/s002850100109.  Google Scholar

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X. Li, J. Ren, S. A. Campbell, G. S. K. Wolkowicz and H. Zhu, How seasonal forcing influences the complexity of a predator-prey system, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018, ) 785. doi: 10.3934/dcdsb.2018043.  Google Scholar

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Y. Lou and R. B. Salako, Dynamics of a parabolic-ODE competition system in heterogeneous environments, P. Am. Math. Soc., 148 (2020), 3025-3038.  doi: 10.1090/proc/14972.  Google Scholar

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R. M. May, Limit cycles in predator-prey communities, Science, 177 (1972), 900–902. doi: 10.1126/science.177.4052.900.  Google Scholar

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A. B. Medvinsky, S. V. Petrovskii, I. A. Tikhonova, H. Malchow and B.-L. Li, Spatiotemporal complexity of plankton and fish dynamics, SIAM Rev., 44 (2002), 311-370.  doi: 10.1137/S0036144502404442.  Google Scholar

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C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Anal. Theor., 48 (2002), 349-362. doi: 10.1016/S0362-546X(00)00189-9.  Google Scholar

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C. V. Pao, Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 198 (1996), 751–779. doi: 10.1006/jmaa.1996.0111.  Google Scholar

[16]

C. V. Pao and W. H. Ruan, Quasilinear parabolic and elliptic systems with mixed quasimonotone functions, J. Differential Equations, 255 (2013), 1515-1553. doi: 10.1016/j.jde.2013.05.015.  Google Scholar

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H. A. Priestley, Introduction to Complex Analysis, OUP Oxford, 2003.  Google Scholar

[18]

J. Ren, L. Yu and S. Siegmund, Bifurcations and chaos in a discrete predator–prey model with Crowley–Martin functional response, Nonlinear Dyn., 90 (2017), 19-41.  doi: 10.1007/s11071-017-3643-6.  Google Scholar

[19]

M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, Am. Nat., 97 (1963), 209-223.  doi: 10.1086/282272.  Google Scholar

[20]

P. Song, Y. Lou and Y. Xiao, A spatial SEIRS reaction-diffusion model in heterogeneous environment, J. Differential Equations, 267 (2019), 5084-5114. doi: 10.1016/j.jde.2019.05.022.  Google Scholar

[21]

C. Tian, L. Zhang and Z. Ling, The stability of a diffusion model of plankton allelopathy with spatio–temporal delays, Nonlinear Anal. Real World Appl., 10 (2009), 2036-2046.  doi: 10.1016/j.nonrwa.2008.03.016.  Google Scholar

[22]

Z.-C. Wang, W.-T. Li and S. G. Ruan, Travelling wave fronts in reaction–diffusion systems with spatio–temporal delays, J. Differential Equations, 222 (2006), 185–232. doi: 10.1016/j.jde.2005.08.010.  Google Scholar

[23]

Y. Wang and J. Shi, Analysis of a reaction-diffusion benthic-drift model with strong Allee effect growth, J. Differential Equations, 269 (2020), 7605–7642. doi: 10.1016/j.jde.2020.05.044.  Google Scholar

[24]

J. Wang, J. Wei and J. Shi, Global bifurcation analysis and pattern formation in homogeneous diffusive predator–prey systems, J. Differential Equations, 260 (2016), 3495-3523.  doi: 10.1016/j.jde.2015.10.036.  Google Scholar

[25]

S. Wu and Y. Song, Stability and spatiotemporal dynamics in a diffusive predator–prey model with nonlocal prey competition, Nonlinear Anal. Real World Appl., 48 (2019), 12-39.  doi: 10.1016/j.nonrwa.2019.01.004.  Google Scholar

[26]

F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator–prey system, J. Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024.  Google Scholar

show all references

References:
[1]

L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst. Ser. A, 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.  Google Scholar

[2]

S. Chen and J. Shi, Asymptotic profiles of basic reproduction number for epidemic spreading in heterogeneous environment, SIAM J. Appl. Math., 80 (2020), 1247-1271.  doi: 10.1137/19M1289078.  Google Scholar

[3]

S. Chen and J. Yu, Stability analysis of a reaction–diffusion equation with spatiotemporal delay and Dirichlet boundary condition, J. Dynam. Differential Equations, 28 (2016), 857-866.  doi: 10.1007/s10884-014-9384-z.  Google Scholar

[4]

K. S. Cheng, Uniqueness of a limit cycle for a predator-prey system, SIAM J. Math. Anal., 12 (1981), 541-548.  doi: 10.1137/0512047.  Google Scholar

[5]

R. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differential Equations, 6 (2016), 3305-3343.  doi: 10.1016/j.jde.2016.05.025.  Google Scholar

[6]

S. A. Gourley and M. V. Bartuccelli, Parameter domains for instability of uniform states in systems with many delays, J. Math. Biol., 35 (1997), 843-867.  doi: 10.1007/s002850050080.  Google Scholar

[7]

S. A. Gourley and N. F. Britton, A predator-prey reaction-diffusion system with nonlocal effects, J. Math. Biol., 34 (1996) 297–333. doi: 10.1007/BF00160498.  Google Scholar

[8]

S. A. Gourley and S. Ruan, Convergence and travelling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. Anal., 35 (2003), 806-822.  doi: 10.1137/S003614100139991.  Google Scholar

[9]

S. A. Gourley and J. W.-H. So, Dynamics of a food-limited population model incorporating nonlocal delays on a finite domain, J. Math. Biol., 44 (2002), 49-78.  doi: 10.1007/s002850100109.  Google Scholar

[10]

X. Li, J. Ren, S. A. Campbell, G. S. K. Wolkowicz and H. Zhu, How seasonal forcing influences the complexity of a predator-prey system, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018, ) 785. doi: 10.3934/dcdsb.2018043.  Google Scholar

[11]

Y. Lou and R. B. Salako, Dynamics of a parabolic-ODE competition system in heterogeneous environments, P. Am. Math. Soc., 148 (2020), 3025-3038.  doi: 10.1090/proc/14972.  Google Scholar

[12]

R. M. May, Limit cycles in predator-prey communities, Science, 177 (1972), 900–902. doi: 10.1126/science.177.4052.900.  Google Scholar

[13]

A. B. Medvinsky, S. V. Petrovskii, I. A. Tikhonova, H. Malchow and B.-L. Li, Spatiotemporal complexity of plankton and fish dynamics, SIAM Rev., 44 (2002), 311-370.  doi: 10.1137/S0036144502404442.  Google Scholar

[14]

C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Anal. Theor., 48 (2002), 349-362. doi: 10.1016/S0362-546X(00)00189-9.  Google Scholar

[15]

C. V. Pao, Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 198 (1996), 751–779. doi: 10.1006/jmaa.1996.0111.  Google Scholar

[16]

C. V. Pao and W. H. Ruan, Quasilinear parabolic and elliptic systems with mixed quasimonotone functions, J. Differential Equations, 255 (2013), 1515-1553. doi: 10.1016/j.jde.2013.05.015.  Google Scholar

[17]

H. A. Priestley, Introduction to Complex Analysis, OUP Oxford, 2003.  Google Scholar

[18]

J. Ren, L. Yu and S. Siegmund, Bifurcations and chaos in a discrete predator–prey model with Crowley–Martin functional response, Nonlinear Dyn., 90 (2017), 19-41.  doi: 10.1007/s11071-017-3643-6.  Google Scholar

[19]

M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, Am. Nat., 97 (1963), 209-223.  doi: 10.1086/282272.  Google Scholar

[20]

P. Song, Y. Lou and Y. Xiao, A spatial SEIRS reaction-diffusion model in heterogeneous environment, J. Differential Equations, 267 (2019), 5084-5114. doi: 10.1016/j.jde.2019.05.022.  Google Scholar

[21]

C. Tian, L. Zhang and Z. Ling, The stability of a diffusion model of plankton allelopathy with spatio–temporal delays, Nonlinear Anal. Real World Appl., 10 (2009), 2036-2046.  doi: 10.1016/j.nonrwa.2008.03.016.  Google Scholar

[22]

Z.-C. Wang, W.-T. Li and S. G. Ruan, Travelling wave fronts in reaction–diffusion systems with spatio–temporal delays, J. Differential Equations, 222 (2006), 185–232. doi: 10.1016/j.jde.2005.08.010.  Google Scholar

[23]

Y. Wang and J. Shi, Analysis of a reaction-diffusion benthic-drift model with strong Allee effect growth, J. Differential Equations, 269 (2020), 7605–7642. doi: 10.1016/j.jde.2020.05.044.  Google Scholar

[24]

J. Wang, J. Wei and J. Shi, Global bifurcation analysis and pattern formation in homogeneous diffusive predator–prey systems, J. Differential Equations, 260 (2016), 3495-3523.  doi: 10.1016/j.jde.2015.10.036.  Google Scholar

[25]

S. Wu and Y. Song, Stability and spatiotemporal dynamics in a diffusive predator–prey model with nonlocal prey competition, Nonlinear Anal. Real World Appl., 48 (2019), 12-39.  doi: 10.1016/j.nonrwa.2019.01.004.  Google Scholar

[26]

F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator–prey system, J. Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024.  Google Scholar

Figure 1.  Some examples of $ f(u) $ and $ g(u) $. (a). Logistic effect (blue), weak Allee effect (red), strong Allee effect (green). (b). Type I function (blue), Type Ⅱ function (red), Type Ⅲ function (green)
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