Abstract
By utilizing the coincidence degree theory and the related continuation theorem, as well as some prior estimates, we investigate the existence and multiplicity of positive periodic solutions of ratio-dependent food chain model with exploited terms. Some sufficient criteria are established for the existence and multiplicity of periodic solutions.
Highlights
1 Introduction The last years have seen very important progress made on Michaelis-Menten type ratiodependent predator-prey model in mathematical ecology literature, which can be roughly stated as that the per capita predator growth rate should be a function of the ratio of prey to predator abundance and usually takes the form mxy mx x (t) = x(r – kx) – c
Ay + x ay + x where x, y stand for prey and predator density, respectively, r, k, a, c, d, m are positive constants that stand for prey intrinsic growth rate, carrying capacity, half-saturation constant, conversion rate, predator’s death rate, and maximal predator growth rate, respectively
3 Conclusions In this paper, with the help of a continuation theorem based on Gaines and Mawhin’s coincidence degree theory, we study the existence and multiplicity of periodic solutions of a ratio-dependent food chain model with exploited term(s)
Summary
We study the following ratio-dependent food chain model with exploited terms in a periodically varying environment because the variation of the environment plays an important role in many biological and ecological systems: x (t) = x(t) r(t) – k(t)x(t) – b (t)y(t) m (t)y(t) + x(t) – h (t), where h , h , h are nonnegative continuous ω-periodic functions representing exploited terms, the other variables and parameters have the same biological meanings as in system If L is a Fredholm mapping of index zero there exist continuous projectors P : X → X and Q : Z → Z such that Im P = Ker L, Im L = Ker Q = Im(I – Q).
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