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Local stability analysis on Lotka‐Volterra predator‐prey models with prey refuge and harvesting
Author(s) -
Chow Christopher,
Hoti Marvin,
Li Chongming,
Lan Kunquan
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5234
Subject(s) - mathematics , predation , boundary (topology) , constant (computer programming) , stability (learning theory) , equilibrium point , stability theory , saddle point , control theory (sociology) , mathematical analysis , nonlinear system , ecology , geometry , computer science , differential equation , physics , control (management) , quantum mechanics , machine learning , artificial intelligence , biology , programming language
We propose a predator‐prey model by incorporating a constant harvesting rate into a Lotka‐Volterra predator‐prey model with prey refuge. All the positive equilibria and the local stability of the proposed model are studied and analyzed by sorting out the intervals of the parameters involved in the model. These intervals of the parameters exhibit the effects on the dynamical behaviors of prey and predators. The emphasis is put on the ranges of the prey refuge constant and harvesting rate. We show that the model has three‐type positive boundary equilibria and one positive interior equilibrium. By using the qualitative theory for planar systems, we show that the three‐type boundary positive equilibria can be saddles, saddle nodes, topological saddles, or stable or unstable nodes, and the interior positive equilibrium is locally asymptotically stable. Under suitable restrictions on the parameters, we prove that the positive interior equilibrium is a stable node. It remains open that under what conditions on the parameters is the positive interior equilibrium a focus.