Dynamics and pattern formation of a diffusive predator–prey model in the presence of toxicity
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In this paper, we develop a diffusivepredator–prey model with toxins under the homogeneous Neumann boundary condition. First, the persistence property and global asymptotic stability of the constant steady states are established. Then by analyzing the associated characteristic equation, we derive explicit conditions for the existence of nonconstant steady states that emerge through steady-state bifurcation from related constant steady states. Furthermore, the existence and nonexistence of nonconstant positive steady states of this model are studied by considering the effect of large diffusivity. Finally, in order to verify our theoretical results, some numerical simulations are also included. These explicit conditions are numerically verified in detail and further compared to those conditions ensuring Turing instability. It is shown that the numerically observed behaviors are in good agreement with the theoretically proposed results. All theoretical analyses and numerical simulations show that toxic substances have a perceptible effect on the system.
KeywordsTuring instability Pattern formation Bifurcation Nonconstant steady state
This study was funded by National Science Foundation of China (Grant Number: 11571170). The study is partially supported by Natural Science Foundation of Jiangsu Province (Grant Number: BK20150420), and also supported by the Startup Foundation for Introducing Talent of NUIST.
Compliance with ethical standards
Conflict of interests
The authors declare that they have no conflict of interest.
- 1.Hallam, T.G., Clark, C.E.: Non-autonomous logistic equations as models of populations in a deteriorating environment. J. Theoret. Biol. 93(2), 303–311 (1981)Google Scholar
- 3.Hallam, T.G., De Luna, J.T.: Effects of toxicants on populations: a qualitative: approach iii. Environmental and food chain pathways. J. Theoret. Biol. 109(3), 411–429 (1984)Google Scholar
- 5.Dubey, B., Hussain, J.: A model for the allelopathic effect on two competing species. Ecol. Modell. 129(2–3), 195–207 (2000)Google Scholar
- 9.Jana, D., Dolai, P., Pal, A.K., Samanta, G.P.: On the stability and Hopf-bifurcation of a multi-delayed competitive population system affected by toxic substances with imprecise biological parameters. Model. Earth Syst. Environ. 2(3), 110 (2016)Google Scholar
- 12.Chattopadhyay, J.: Effect of toxic substances on a two-species competitive system. Ecol. Modell. 84(1–3), 287–289 (1996)Google Scholar
- 13.Kar, T.K., Chaudhuri, K.S.: On non-selective harvesting of two competing fish species in the presence of toxicity. Ecol. Modell. 161(1–2), 125–137 (2003)Google Scholar
- 15.Jianhong, W.: Theory and Applications of Partial Functional Differential Equations. Applied Mathematical Sciences. Springer, New York (1996)Google Scholar
- 23.Wang, W., Lin, Y., Rao, F., Zhang, L., Tan, Y.: Pattern selection in a ratio-dependent predator-prey model. J. Stat. Mech. Theory and Exp. 2010(11), P11036 (2010)Google Scholar
- 25.Ghorai, S., Poria, S.: Pattern formation in a system involving prey-predation, competition and commensalism. Nonlinear Dyn. 89(2), 1309–1326 (2017)Google Scholar
- 29.Allen, J.C., Schaffer, W.M., Rosko, D.: Chaos reduces species extinction by amplifying local population noise. Nature 364(6434), 229–32 (1993)Google Scholar
- 30.Heino, M., Kaitala, V., Ranta, E., Lindstrom, J.: Synchronous dynamics and rates of extinction in spatially structured populations. Proc. R. Soc. B Biol. Sci. 264(1381), 481–486 (1997)Google Scholar
- 31.Schimanskygeier, L., Fiedler, B., Kurths, J., Scholl, E.: Analysis and Control of Complex Nonlinear Processes in Physics. Chemistry and Biology. World Scientific, Singapore (2007)Google Scholar