Abstract
The main concern of this paper is to study the dynamic of a predator–prey system with diffusion. It incorporates the Holling-type-II and a modified Leslie–Gower functional responses under Robin boundary conditions. More concretely, we study the dissipativeness of the system by using the comparison principle, and we derive a criteria for permanence and for predator extinction.
Similar content being viewed by others
References
Abid W, Yafia R, Aziz-Alaoui MA, Bouhafa H, Abichou A (2015) Instability and pattern formation in three-species food chain model via Holling type II functional response on a circular domain. Int J Bifurc Chaos 25(6):1550092
Abid W, Yafia R, Aziz-Alaoui MA, Bouhafa H, Abichou A (2015) Diffusion driven instability and Hopf bifurcation in spatial predator–prey model on a circular domain. Appl Math Comput 260:292–313
Aziz-Alaoui MA (2002) Study of a Leslie–Gower-type tritrophic population model. Chaos Solitons Fractals 14(8):1275–1293
Aziz-Alaoui MA, Daher OM (2003) Boudedness and global stability for a predator–prey model with modified Leslie–Gower and Holling type II schemes. Appl Math Lett 16(7):1069–1075
Bassett A, Krause A, Van Gorder RA (2017) Continuous dispersal in a model of predator–prey-subsidy population dynamics. Ecol Model 354:115–122
Camara BI, Aziz-Alaoui MA (2008) Dynamics of a predator–prey model with diffusion. Dyn Contin Discrete Impuls Syst Ser A Math Anal 15:897–906
Camara BI, Aziz-Alaoui MA (2008) Complexity in a prey predator model. ARIMA 9:109–122
Cantrell RS, Cosner C (1991) Diffusive logistic equations with indefinite weights: population models in disrupted environments II. SIAM J Math Anal 22:1043–1064
Cantrell RS, Cosner C (1996) Practical persistence in ecological models via comparison methods. Proc R Soc Edinb Sect A 126:247–272
Cantrell RS, Cosner C (1998) Practical persistence in diffusive food chain models. Nat Res Model 11:21–34
Cantrell RS, Cosner C (1999) Diffusion models for population dynamics incorporating individual behavior at boundaries: applications to refuge design. Theor Popul Biol 55:189–207
Cantrell RS, Cosner C (2001) On the dynamics of predator–prey models with the Beddington–DeAngelis functional response. J Math Anal Appl 257:206–222
Cantrell RS, Cosner C (2003) Spatial ecology via reaction–diffusion equations. Wiley Ser Math Comput Biol. Wiley, Chichester
Chen S, Shi J (2012) Global stability in a diffusive Holling–Tanner predator–prey model. Appl Math Lett 25(3):614–618
Chen B, Wang M (2008) Qualitative analysis for a diffusive predator–prey model. Comput Math Appl 55:339–355
Daher MO (2004) Etude et analyse asymptotique de certains systèmes dynamiques non-linéaires : application à des problèmes proie-prédateurs (in french). PhD Thesis, University of Le Havre, France
Daher Okiye M, Aziz-Alaoui MA (2003) On the dynamics of a predator-prey model with the Holling–Tanner functional response. In: Mathematical modelling and computing in biology and medicine, Milan Res Cent Ind Appl Math MIRIAM Proj, Esculapio, Bologna, 92D25, pp 270–278
Dai G, Ma R, Wang H, Wang F, Xu K (2015) Partial differential equations with Robin boundary conditions in online social networks. Discrete Contin Dyn Syst B 20(6):1609–1624
Friedman A (1964) Partial differential equations of parabolic type. Prentice-Hall Inc, Englewood Cliffs
Hale JK (1988) Asymptotic behavior of dissipative systems. Math Surveys and Monographs, vol 25. Am Math Sot, Providence, RI
Hale JK, Waltman P (1989) Persistence in infinite-dimensional systems. SIAM J Math Anal 20:388–395
Hess P (1991) Periodic-parabolic boundary value problems and positivity. Longman Scientific & Technical, Harlow, Essex.
Hutson V, Schmitt K (1992) Permanence in dynamical systems. Math Biosci 111:1–17
Ko W, Ryu K (2006) Qualitative analysis of a predator–prey model with Holling type II functional response incorporating a prey refuge. J Differ Equ 231(2):534–550
Kurowski L, Krause A, Mizuguchi H, Grindrod P, Van Gorder RA (2017) Two-species migration and clustering in two-dimensional domains. Bull Math Biol 79(10):2302–2333
Letellier C, Aziz-Alaoui MA (2002) Analysis of the dynamics of a realistic ecological model. Chaos Solitons Fractals 13(1):95–107
Letellier C, Aguirre L, Maquet J, Aziz-Alaoui MA (2002) Should all the species of a food chain be counted to investigate the global dynamics. Chaos Solitons Fractals 13(5):1099–1113
Moussaoui A, Bouguima SM (2016) Seasonal influences on a preypredator model. J Appl Math Comput 50(1):39–57
Nindjin AF, Aziz-Alaoui MA (2008) Persistence and global stability in a delayed Leslie–Gower type three species food chain. J Math Anal Appl 340(1):340–357
Nindjin AF, Aziz-Alaoui MA, Cadivel M (2006) Analysis of a predator–prey model with modified Leslie–Gower and Holling-type II schemes with time delay. Nonlinear Anal Real World Appl 7:1104–1118
Pao CV (1982) On nonlinear reaction–diffusion systems. J Math Anal Appl 87(1):165–198
Saez E, Gonzalez-Olivares E (1999) Dynamics of a predator–prey model. SIAM J Appl Math 59(5):1867–1878
Singh A, Gakkhar S (2014) Stabilization of modified Leslie–Gower prey–predator model. Differ Equ Dyn Syst 22(3):239–249
Tanner JT (1975) The stability and the intrinsic growth rates of prey and predator populations. Ecology 56:855–867
Upadhyay RK, Kumari N, Vikas Rai V (2008) Wave of chaos and pattern formation in spatial predator–prey systems with Holling type IV predator response. Math Model Nat Phenom 3(4):71–95
Upadhyay RK, Kumari N, Vikas Rai V (2009) Wave phenomena and edge of chaos in a diffusive predator–prey system under Allee effect. Differ Equ Dyn Syst 17(3):301–317
Wang F, Wang H, Xu K (2012) Diffusive logistic model towards predicting information diffusion in online social networks. In: 32nd international conference on distributed computing systems workshops (ICDCSW), pp 133–139
Yafia R, El Adnani F, Talibi H (2007) Stability of limit cycle in a predator-prey model with modified Leslie–Gower and Holling-type II schemes with time delay. Appl Math Sci 1(3):119–131
Yafia R, El Adnani F, Talibi H (2008) Limit cycle and numerical similations for small and large delays in a predatorprey model with modified Leslie–Gower and Holling-type II scheme. Nonlinear Anal Real World Appl 9:2055–2067
Ye QX, Li ZY (1990) Introduction to reaction–diffusion equations. Science Press, Beijing
Acknowledgements
This work was partially supported by the ERDF (XTerm Project) and Normandie Region, France. And also by The LMAH-FR-CNRS-3335.
Author information
Authors and Affiliations
Corresponding author
Appendix: Permanence
Appendix: Permanence
For the convenience of the reader, we will summarize some facts contained in Hale and Waltman, see Hale and Waltman (1989), about the permanence for abstract dynamical systems. Suppose that \(\varOmega\) is a complete metric space with \(\varOmega =\varOmega _0\cup \partial \varOmega _0\) for an open set \(\varOmega _0\), where \(\partial \varOmega _0\) is the boundary of the set \(\varOmega _0\). We will typically choose \(\varOmega _0\) to be the positive cone in an ordered Banach space. A flow or semiflow on \(\varOmega\) under which \(\varOmega _0\) and \(\partial \varOmega _0\) are forward invariant is said to be permanent if it is dissipative and if there is a number \(\eta >0\) such that any trajectory starting in \(\varOmega _0\) will be at least a distance \(\eta\) from \(\partial \varOmega _0\) for all sufficiently large t. To state a theorem implying permanence we need a few definitions. An invariant set M for the flow or semiflow is said to be isolated if it has a neighborhood U such that M is the maximal invariant subset of U. Let \(\omega (\partial \varOmega _0) \subset \partial \varOmega _0\) denote the union of the sets \(\omega (u)\) over \(u\in \varOmega _0\) (This differs from the standard definition of the \(\omega\)-limit set of a set but it is more convenient for our purposes; see Hutson and Schmitt (1992) for a discussion). The set \(\omega (\varOmega _0)\) is said to be isolated if it has a covering \(M=\cup _{k=1}^{N}M_{k}\) of pairwise disjoint, both sets \(M_k\) which are isolated and invariant with respect to the flow or semiflow both on \(\partial \varOmega _0\) and on \(\varOmega =\varOmega _0\cup \partial \varOmega _0\). The covering M is then called an isolated covering. Suppose \(N_1\) and \(N_2\) are isolated invariant sets (not necessarily distinct). The set \(N_1\) is said to be chained to \(N_2\) (denoted \(N_{1}\rightarrow N_{2}\)) if there exists \(u\in N_{1}\cup N_{2}\) with \(u\in {W}^{u} (N_{1})\cap {W}^{s}(N_{2}).\) (As usual, \(W^{u}\) and \(W^{s}\) denote the unstable and stable manifolds, respectively). A finite sequence \(N_{1}, N_{2},\cdots , N_{k}\) of isolated invariant sets is a chain if \(N_{1}\rightarrow N_{2}\rightarrow N_{3}\rightarrow \cdots \rightarrow N_{k}\). (This is possible for \(k = 1\) if \(N_{1}\rightarrow N_{1}.)\) The chain is called a cycle if \(N_{k}= N_{1}.\)The set \(\omega (\partial \varOmega _{0})\) is said to be acyclic if there exists an isolated covering \(\cup _{k=1}^{N}M_{k}\) such that no subset of \(\left\{ M_{k}\right\}\) is a cycle. We now state a theorem that can be used to establish permanence.
Theorem 3
(Hale and Waltman 1989) Suppose that \(\varOmega\) is a complete metric space with \(\varOmega =\varOmega _0\cup \partial \varOmega _0\) where \(\varOmega _0\) is open. Suppose that a semiflow on \(\varOmega\) leaves both \(\varOmega _{0}\) and \(\partial \varOmega _{0}\) forward invariant, maps bounded sets in \(\varOmega\) to precompact set for \(t>0\), and is dissipative. If in addition:
-
(i)
\(\omega (\partial \varOmega _{0})\) is isolated and acyclic,
-
(ii)
\(W^{s}(M_{k})\cap \varOmega _{0}=\emptyset\) for all k, where \(\cup _{k=1}^{N}M_{k}\) is the isolated covering used in the definition of acyclicity of \(\partial \varOmega _{0}\), then the semiflow is permanent; i.e., there exist \(\eta >0\) such that any trajectory with initial data in \(\varOmega _{0}\) will be bounded away from \(\partial \varOmega _{0}\) by a distance greater than \(\eta\) for t sufficiently large.
Rights and permissions
About this article
Cite this article
Aziz-Alaoui, M.A., Daher Okiye, M. & Moussaoui, A. Permanence and Extinction of a Diffusive Predator–Prey Model with Robin Boundary Conditions. Acta Biotheor 66, 367–378 (2018). https://doi.org/10.1007/s10441-018-9332-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10441-018-9332-0