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doi: 10.3934/dcdss.2020132

Spatiotemporal dynamics of a diffusive predator-prey model with generalist predator

1. 

School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

2. 

Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China

3. 

Science and Mathematics Faculty, College of Integrative Sciences and Arts, Arizona State University, Mesa, AZ 85212, USA

4. 

Simon A. Levin Mathematical, Computational, and Modeling Sciences Center, Arizona State University, Tempe, AZ 85281, USA

* Corresponding author: Jianshe Yu

Received  December 2018 Revised  February 2019 Published  November 2019

In this paper, we study the spatiotemporal dynamics of a diffusive predator-prey model with generalist predator subject to homogeneous Neumann boundary condition. Some basic dynamics including the dissipation, persistence and non-persistence(i.e., one species goes extinct), the local and global stability of non-negative constant steady states of the model are investigated. The conditions of Turing instability due to diffusion at positive constant steady states are presented. A critical value $ \rho $ of the ratio $ \frac{d_2}{d_1} $ of diffusions of predator to prey is obtained, such that if $ \frac{d_2}{d_1}>\rho $, then along with other suitable conditions Turing bifurcation will emerge at a positive steady state, in particular so it is with the large diffusion rate of predator or the small diffusion rate of prey; while if $ \frac{d_2}{d_1}<\rho $, both the reaction-diffusion system and its corresponding ODE system are stable at the positive steady state. In addition, we provide some results on the existence and non-existence of positive non-constant steady states. These existence results indicate that the occurrence of Turing bifurcation, along with other suitable conditions, implies the existence of non-constant positive steady states bifurcating from the constant solution. At last, by numerical simulations, we demonstrate Turing pattern formation on the effect of the varied diffusive ratio $ \frac{d_2}{d_1} $. As $ \frac{d_2}{d_1} $ increases, Turing patterns change from spots pattern, stripes pattern into spots-stripes pattern. It indicates that the pattern formation of the model is rich and complex.

Citation: Dingyong Bai, Jianshe Yu, Yun Kang. Spatiotemporal dynamics of a diffusive predator-prey model with generalist predator. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020132
References:
[1]

J. Blat and K. J. Brown, Global bifurcation of positive solutions in some systems of elliptic equation, SIAM J. Math. Anal., 17 (1986), 1339-1353.  doi: 10.1137/0517094.  Google Scholar

[2]

J. L. BrownV. Morales and K. Summers, Tactical reproductive parasitism via larval cannibalism in Peruvian poison frogs, Biol. Lett., 5 (2009), 148-151.  doi: 10.1098/rsbl.2008.0591.  Google Scholar

[3]

Y. L. CaiM. BanerjeeY. Kang and W. M. Wang, Spationtemporal complexity in a predator-prey model with weak Allee effects, Math. Biosci. Eng., 11 (2014), 1247-1274.  doi: 10.3934/mbe.2014.11.1247.  Google Scholar

[4]

Y. L. Cai, Z. J. Gui, X. B. Zhang, H. B. Shi and W. M. Wang, Bifurcations and pattern formation in a predator-prey model, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850140, 17 pp. doi: 10.1142/S0218127418501407.  Google Scholar

[5]

S. Chakraborty, The infuence of generalist predators in spatially extended predator-prey systems, Ecological Complexity, 23 (2015), 50-60.  doi: 10.1016/j.ecocom.2015.06.003.  Google Scholar

[6]

D. W. Crowder and W. E. Snyder, Eating their way to the top? Mechanisms underlying the success of invasive generalist predators, Biol. Invas., 12 (2010), 2857-2876.  doi: 10.1007/s10530-010-9733-8.  Google Scholar

[7]

Y. H. Du and J. P. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Tran. Amer. Math. Soc., 359 (2007), 4557-4593.  doi: 10.1090/S0002-9947-07-04262-6.  Google Scholar

[8]

Y. H. Du and J. P. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations, 229 (2006), 63-91.  doi: 10.1016/j.jde.2006.01.013.  Google Scholar

[9]

A. ErbachF. Lutscher and G. Seo, Bistability and limit cycles in generalist predator-prey dynamics, Ecological Complexity, 14 (2014), 48-55.  doi: 10.1016/j.ecocom.2013.02.005.  Google Scholar

[10]

Y.-H. Fan and W.-T. Li, Global asymptotic stability of a ratio-dependent predator-prey system with diffusion, J. Comput. Appl. Math., 188 (2006), 205-227.  doi: 10.1016/j.cam.2005.04.007.  Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition, Grundlehren der Mathematischen Wissenschaften, 224. Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-96379-7.  Google Scholar

[12]

R. J. Han and B. X. Dai, Spatiotemporal pattern formation and selection induced by nonlinear cross-diffusion in a toxic-phytoplankton-zooplankton model with Allee effect, Nonlinear Anal. Real World Appl., 45 (2019), 822-853.  doi: 10.1016/j.nonrwa.2018.05.018.  Google Scholar

[13]

I. HanskiL. Hansson and H. Henttonen, Specialist predators, generalist predators, and the microtine rodent cycle, J. Anim. Ecol., 60 (1991), 353-367.  doi: 10.2307/5465.  Google Scholar

[14] M. P. Hassell, The Dynamics of Arthropod Predator-Prey Systems, Monographs in Population Biology, 13. Princeton University Press, Princeton, N.J., 1978.  doi: 10.2307/3280305.  Google Scholar
[15]

X. He and S. N. Zheng, Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response, J. Math. Biol., 75 (2017), 239-257.  doi: 10.1007/s00285-016-1082-5.  Google Scholar

[16]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. doi: 10.1007/BFb0089649.  Google Scholar

[17]

T. Kadota and K. Kuto, Positive steady states for a prey-predator model with some nonlinear diffusion terms, J. Math. Anal. Appl., 323 (2006), 1387-1401.  doi: 10.1016/j.jmaa.2005.11.065.  Google Scholar

[18]

Y. Kang and J. H. Fewell, Co-evolutionary dynamics of a social parasite-host interaction model: obligate versus facultative social parasitism, Nat. Resour. Model., 28 (2015), 398-455.  doi: 10.1111/nrm.12078.  Google Scholar

[19]

Y. Kang and L. Wedekin, Dynamics of a intraguild predation model with generalist or specialist predator, J. Math. Biol., 67 (2013), 1227-1259.  doi: 10.1007/s00285-012-0584-z.  Google Scholar

[20]

X. L. Li, G. P. Hu and Z. S. Feng, Pattern dynamics in a spatial predator-prey model with nonmonotonic response function, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850077, 17 pp. doi: 10.1142/S0218127418500773.  Google Scholar

[21]

C.-S. LinW.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.  Google Scholar

[22]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.  Google Scholar

[23]

S. MadecJ. CasasG. Barles and C. Suppo, Bistability induced by generalist natural enemies can reverse pest invasions, J. Math. Biol., 75 (2017), 543-575.  doi: 10.1007/s00285-017-1093-x.  Google Scholar

[24]

C. MagalC. CosnerS. G. Ruan and J. Casas, Control of invasive hosts by generalist parasitoids, Math. Medi. Biol., 25 (2008), 1-20.  doi: 10.1093/imammb/dqm011.  Google Scholar

[25]

Z. Mei, Numerical Bifurcation Analysis for Reaction-Diffusion Equations, Springer Series in Computational Mathematics, 28. Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-662-04177-2.  Google Scholar

[26]

P. Y. H. Pang and M. X. Wang, Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 919-942.  doi: 10.1017/S0308210500002742.  Google Scholar

[27]

R. Peng and J. P. Shi, Non-existence of non-constant positive steady states of two Holling type-Ⅱ predator-prey systems: Strong interaction case, J. Differential Equations, 247 (2009), 866-886.  doi: 10.1016/j.jde.2009.03.008.  Google Scholar

[28]

R. PengM. X. Wang and M. Yang, Positive solutions of a diffusive prey-predator model in a heterogeneous environment, Math. Comput. Modelling, 46 (2007), 1410-1418.  doi: 10.1016/j.mcm.2007.02.001.  Google Scholar

[29]

J. A. RosenheimH. K. KayaL. E. EhlerJ. J. Marois and B. A. Jaffee, Intraguild predation among biological control agents: Theory and evidence, Biol. Control, 5 (1995), 303-335.  doi: 10.1006/bcon.1995.1038.  Google Scholar

[30]

S. I. Rothstein, Evolutionary rates and host defenses against avian brood parasitism, The American Naturalist, 109 (1975), 161-176.  doi: 10.1086/282984.  Google Scholar

[31]

W. E. Snyder and A. R. Ives, Interactions between specialist and generalist natural enemies: parasitoids, predators, and pea aphid biocontrol, Ecology, 84 (2003), 91-107.   Google Scholar

[32]

Y. L. SongH. P. JiangQ.-X. Liu and Y. Yuan, Spatiotemporal dynamics of the diffusive Mussel-Algae model near Turing-Hopf bifurcation, SIAM J. Appl. Dyn. Syst., 16 (2017), 2030-2062.  doi: 10.1137/16M1097560.  Google Scholar

[33]

M. D. Sorenson, Effects of intra- and interspecific brood parasitism on a precocial host, the canvasback, Aythya valisineria, Behavioral Ecology, 8 (1997), 153-161.  doi: 10.1093/beheco/8.2.153.  Google Scholar

[34] C. N. SpottiswoodeR. M. Kilner and N. B. Davies, Brood Parasitism, The Evolution of Parental Care, Oxford University Press, Oxford, 2012.  doi: 10.1093/acprof:oso/9780199692576.003.0001.  Google Scholar
[35]

W. O. C. SymondsonK. D. Sunderland and M. H. Greenstone, Can generalist predators be effective biocontrol agents?, Annu. Rev. Entomol., 47 (2002), 561-594.  doi: 10.1146/annurev.ento.47.091201.145240.  Google Scholar

[36]

Q. WangC. Y. Gai and J. D. Yan, Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239-1284.  doi: 10.3934/dcds.2015.35.1239.  Google Scholar

[37]

W. M. WangX. Y. GaoY. L. CaiH. B. Shi and S. M. Fu, Turing patterns in a diffusive epidemic model with saturated infection force, J. Franklin Inst., 355 (2018), 7226-7245.  doi: 10.1016/j.jfranklin.2018.07.014.  Google Scholar

[38]

S. H. Wu and Y. L. 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

show all references

References:
[1]

J. Blat and K. J. Brown, Global bifurcation of positive solutions in some systems of elliptic equation, SIAM J. Math. Anal., 17 (1986), 1339-1353.  doi: 10.1137/0517094.  Google Scholar

[2]

J. L. BrownV. Morales and K. Summers, Tactical reproductive parasitism via larval cannibalism in Peruvian poison frogs, Biol. Lett., 5 (2009), 148-151.  doi: 10.1098/rsbl.2008.0591.  Google Scholar

[3]

Y. L. CaiM. BanerjeeY. Kang and W. M. Wang, Spationtemporal complexity in a predator-prey model with weak Allee effects, Math. Biosci. Eng., 11 (2014), 1247-1274.  doi: 10.3934/mbe.2014.11.1247.  Google Scholar

[4]

Y. L. Cai, Z. J. Gui, X. B. Zhang, H. B. Shi and W. M. Wang, Bifurcations and pattern formation in a predator-prey model, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850140, 17 pp. doi: 10.1142/S0218127418501407.  Google Scholar

[5]

S. Chakraborty, The infuence of generalist predators in spatially extended predator-prey systems, Ecological Complexity, 23 (2015), 50-60.  doi: 10.1016/j.ecocom.2015.06.003.  Google Scholar

[6]

D. W. Crowder and W. E. Snyder, Eating their way to the top? Mechanisms underlying the success of invasive generalist predators, Biol. Invas., 12 (2010), 2857-2876.  doi: 10.1007/s10530-010-9733-8.  Google Scholar

[7]

Y. H. Du and J. P. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Tran. Amer. Math. Soc., 359 (2007), 4557-4593.  doi: 10.1090/S0002-9947-07-04262-6.  Google Scholar

[8]

Y. H. Du and J. P. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations, 229 (2006), 63-91.  doi: 10.1016/j.jde.2006.01.013.  Google Scholar

[9]

A. ErbachF. Lutscher and G. Seo, Bistability and limit cycles in generalist predator-prey dynamics, Ecological Complexity, 14 (2014), 48-55.  doi: 10.1016/j.ecocom.2013.02.005.  Google Scholar

[10]

Y.-H. Fan and W.-T. Li, Global asymptotic stability of a ratio-dependent predator-prey system with diffusion, J. Comput. Appl. Math., 188 (2006), 205-227.  doi: 10.1016/j.cam.2005.04.007.  Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition, Grundlehren der Mathematischen Wissenschaften, 224. Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-96379-7.  Google Scholar

[12]

R. J. Han and B. X. Dai, Spatiotemporal pattern formation and selection induced by nonlinear cross-diffusion in a toxic-phytoplankton-zooplankton model with Allee effect, Nonlinear Anal. Real World Appl., 45 (2019), 822-853.  doi: 10.1016/j.nonrwa.2018.05.018.  Google Scholar

[13]

I. HanskiL. Hansson and H. Henttonen, Specialist predators, generalist predators, and the microtine rodent cycle, J. Anim. Ecol., 60 (1991), 353-367.  doi: 10.2307/5465.  Google Scholar

[14] M. P. Hassell, The Dynamics of Arthropod Predator-Prey Systems, Monographs in Population Biology, 13. Princeton University Press, Princeton, N.J., 1978.  doi: 10.2307/3280305.  Google Scholar
[15]

X. He and S. N. Zheng, Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response, J. Math. Biol., 75 (2017), 239-257.  doi: 10.1007/s00285-016-1082-5.  Google Scholar

[16]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. doi: 10.1007/BFb0089649.  Google Scholar

[17]

T. Kadota and K. Kuto, Positive steady states for a prey-predator model with some nonlinear diffusion terms, J. Math. Anal. Appl., 323 (2006), 1387-1401.  doi: 10.1016/j.jmaa.2005.11.065.  Google Scholar

[18]

Y. Kang and J. H. Fewell, Co-evolutionary dynamics of a social parasite-host interaction model: obligate versus facultative social parasitism, Nat. Resour. Model., 28 (2015), 398-455.  doi: 10.1111/nrm.12078.  Google Scholar

[19]

Y. Kang and L. Wedekin, Dynamics of a intraguild predation model with generalist or specialist predator, J. Math. Biol., 67 (2013), 1227-1259.  doi: 10.1007/s00285-012-0584-z.  Google Scholar

[20]

X. L. Li, G. P. Hu and Z. S. Feng, Pattern dynamics in a spatial predator-prey model with nonmonotonic response function, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850077, 17 pp. doi: 10.1142/S0218127418500773.  Google Scholar

[21]

C.-S. LinW.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.  Google Scholar

[22]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.  Google Scholar

[23]

S. MadecJ. CasasG. Barles and C. Suppo, Bistability induced by generalist natural enemies can reverse pest invasions, J. Math. Biol., 75 (2017), 543-575.  doi: 10.1007/s00285-017-1093-x.  Google Scholar

[24]

C. MagalC. CosnerS. G. Ruan and J. Casas, Control of invasive hosts by generalist parasitoids, Math. Medi. Biol., 25 (2008), 1-20.  doi: 10.1093/imammb/dqm011.  Google Scholar

[25]

Z. Mei, Numerical Bifurcation Analysis for Reaction-Diffusion Equations, Springer Series in Computational Mathematics, 28. Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-662-04177-2.  Google Scholar

[26]

P. Y. H. Pang and M. X. Wang, Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 919-942.  doi: 10.1017/S0308210500002742.  Google Scholar

[27]

R. Peng and J. P. Shi, Non-existence of non-constant positive steady states of two Holling type-Ⅱ predator-prey systems: Strong interaction case, J. Differential Equations, 247 (2009), 866-886.  doi: 10.1016/j.jde.2009.03.008.  Google Scholar

[28]

R. PengM. X. Wang and M. Yang, Positive solutions of a diffusive prey-predator model in a heterogeneous environment, Math. Comput. Modelling, 46 (2007), 1410-1418.  doi: 10.1016/j.mcm.2007.02.001.  Google Scholar

[29]

J. A. RosenheimH. K. KayaL. E. EhlerJ. J. Marois and B. A. Jaffee, Intraguild predation among biological control agents: Theory and evidence, Biol. Control, 5 (1995), 303-335.  doi: 10.1006/bcon.1995.1038.  Google Scholar

[30]

S. I. Rothstein, Evolutionary rates and host defenses against avian brood parasitism, The American Naturalist, 109 (1975), 161-176.  doi: 10.1086/282984.  Google Scholar

[31]

W. E. Snyder and A. R. Ives, Interactions between specialist and generalist natural enemies: parasitoids, predators, and pea aphid biocontrol, Ecology, 84 (2003), 91-107.   Google Scholar

[32]

Y. L. SongH. P. JiangQ.-X. Liu and Y. Yuan, Spatiotemporal dynamics of the diffusive Mussel-Algae model near Turing-Hopf bifurcation, SIAM J. Appl. Dyn. Syst., 16 (2017), 2030-2062.  doi: 10.1137/16M1097560.  Google Scholar

[33]

M. D. Sorenson, Effects of intra- and interspecific brood parasitism on a precocial host, the canvasback, Aythya valisineria, Behavioral Ecology, 8 (1997), 153-161.  doi: 10.1093/beheco/8.2.153.  Google Scholar

[34] C. N. SpottiswoodeR. M. Kilner and N. B. Davies, Brood Parasitism, The Evolution of Parental Care, Oxford University Press, Oxford, 2012.  doi: 10.1093/acprof:oso/9780199692576.003.0001.  Google Scholar
[35]

W. O. C. SymondsonK. D. Sunderland and M. H. Greenstone, Can generalist predators be effective biocontrol agents?, Annu. Rev. Entomol., 47 (2002), 561-594.  doi: 10.1146/annurev.ento.47.091201.145240.  Google Scholar

[36]

Q. WangC. Y. Gai and J. D. Yan, Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239-1284.  doi: 10.3934/dcds.2015.35.1239.  Google Scholar

[37]

W. M. WangX. Y. GaoY. L. CaiH. B. Shi and S. M. Fu, Turing patterns in a diffusive epidemic model with saturated infection force, J. Franklin Inst., 355 (2018), 7226-7245.  doi: 10.1016/j.jfranklin.2018.07.014.  Google Scholar

[38]

S. H. Wu and Y. L. 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

Figure 1.  Stationary hot spots pattern in model (2). The parameter values are taken as (32) and $ (d_1, d_2) = (0.028, 0.226) $. The zero-flux boundary condition is used and initial condition is small perturbation around the homogeneous steady-state $ E^* = (1.5934, 2.0768) $.
Figure 2.  Stationary stripes pattern in model (2). The parameter values are taken as (32) and $ (d_1, d_2) = (0.028, 0.27) $. The zero-flux boundary condition is used and initial condition is small perturbation around the homogeneous steady-state $ E^* = (1.5934, 2.0768) $.
Figure 3.  Stationary spots-stripes pattern in model (2). The parameter values are taken as (32) and $ (d_1, d_2) = (0.028, 0.45) $. The zero-flux boundary condition is used and initial condition is small perturbation around the homogeneous steady-state $ E^* = (1.5934, 2.0768) $
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