# Permanence and Extinction of a Diffusive Predator–Prey Model with Robin Boundary Conditions

Regular Article

First Online:

- 113 Downloads

## 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.

## Keywords

Reaction–diffusion Robin boundary conditions Predator–prey Permanence Extinction## Mathematics Subject Classification

35B40 37N25## Notes

### Acknowledgements

This work was partially supported by the ERDF (XTerm Project) and Normandie Region, France. And also by The LMAH-FR-CNRS-3335.

## 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):1550092CrossRefGoogle Scholar
- 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 260:292–313Google Scholar
- Aziz-Alaoui MA (2002) Study of a Leslie–Gower-type tritrophic population model. Chaos Solitons Fractals 14(8):1275–1293CrossRefGoogle Scholar
- 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):1069–1075CrossRefGoogle Scholar
- Bassett A, Krause A, Van Gorder RA (2017) Continuous dispersal in a model of predator–prey-subsidy population dynamics. Ecol Model 354:115–122CrossRefGoogle Scholar
- Camara BI, Aziz-Alaoui MA (2008) Dynamics of a predator–prey model with diffusion. Dyn Contin Discrete Impuls Syst Ser A Math Anal 15:897–906Google Scholar
- Camara BI, Aziz-Alaoui MA (2008) Complexity in a prey predator model. ARIMA 9:109–122Google Scholar
- Cantrell RS, Cosner C (1991) Diffusive logistic equations with indefinite weights: population models in disrupted environments II. SIAM J Math Anal 22:1043–1064CrossRefGoogle Scholar
- Cantrell RS, Cosner C (1996) Practical persistence in ecological models via comparison methods. Proc R Soc Edinb Sect A 126:247–272CrossRefGoogle Scholar
- Cantrell RS, Cosner C (1998) Practical persistence in diffusive food chain models. Nat Res Model 11:21–34CrossRefGoogle Scholar
- Cantrell RS, Cosner C (1999) Diffusion models for population dynamics incorporating individual behavior at boundaries: applications to refuge design. Theor Popul Biol 55:189–207CrossRefGoogle Scholar
- Cantrell RS, Cosner C (2001) On the dynamics of predator–prey models with the Beddington–DeAngelis functional response. J Math Anal Appl 257:206–222CrossRefGoogle Scholar
- Cantrell RS, Cosner C (2003) Spatial ecology via reaction–diffusion equations. Wiley Ser Math Comput Biol. Wiley, ChichesterGoogle Scholar
- Chen S, Shi J (2012) Global stability in a diffusive Holling–Tanner predator–prey model. Appl Math Lett 25(3):614–618CrossRefGoogle Scholar
- Chen B, Wang M (2008) Qualitative analysis for a diffusive predator–prey model. Comput Math Appl 55:339–355CrossRefGoogle Scholar
- 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, FranceGoogle Scholar
- 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 270–278Google Scholar
- 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):1609–1624CrossRefGoogle Scholar
- Friedman A (1964) Partial differential equations of parabolic type. Prentice-Hall Inc, Englewood CliffsGoogle Scholar
- Hale JK (1988) Asymptotic behavior of dissipative systems. Math Surveys and Monographs, vol 25. Am Math Sot, Providence, RIGoogle Scholar
- Hale JK, Waltman P (1989) Persistence in infinite-dimensional systems. SIAM J Math Anal 20:388–395CrossRefGoogle Scholar
- Hess P (1991) Periodic-parabolic boundary value problems and positivity. Longman Scientific & Technical, Harlow, Essex.Google Scholar
- Hutson V, Schmitt K (1992) Permanence in dynamical systems. Math Biosci 111:1–17CrossRefGoogle Scholar
- 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):534–550CrossRefGoogle Scholar
- 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):2302–2333CrossRefGoogle Scholar
- Letellier C, Aziz-Alaoui MA (2002) Analysis of the dynamics of a realistic ecological model. Chaos Solitons Fractals 13(1):95–107CrossRefGoogle Scholar
- 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):1099–1113CrossRefGoogle Scholar
- Moussaoui A, Bouguima SM (2016) Seasonal influences on a preypredator model. J Appl Math Comput 50(1):39–57CrossRefGoogle Scholar
- 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):340–357CrossRefGoogle Scholar
- 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 7:1104–1118CrossRefGoogle Scholar
- Pao CV (1982) On nonlinear reaction–diffusion systems. J Math Anal Appl 87(1):165–198CrossRefGoogle Scholar
- Saez E, Gonzalez-Olivares E (1999) Dynamics of a predator–prey model. SIAM J Appl Math 59(5):1867–1878CrossRefGoogle Scholar
- Singh A, Gakkhar S (2014) Stabilization of modified Leslie–Gower prey–predator model. Differ Equ Dyn Syst 22(3):239–249CrossRefGoogle Scholar
- Tanner JT (1975) The stability and the intrinsic growth rates of prey and predator populations. Ecology 56:855–867CrossRefGoogle Scholar
- 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):71–95CrossRefGoogle Scholar
- 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):301–317CrossRefGoogle Scholar
- 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 133–139Google Scholar
- 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):119–131Google Scholar
- 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 9:2055–2067CrossRefGoogle Scholar
- Ye QX, Li ZY (1990) Introduction to reaction–diffusion equations. Science Press, BeijingGoogle Scholar

## Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018