Existence, uniqueness and stability analysis of allelopathic stimulatory phytoplankton model (original) (raw)
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In this paper, we investigate a non-autonomous competitive phytoplankton model with periodic coefficients in deterministic and stochastic environment, respectively. We present a detailed mathematical analysis of the deterministic model like uniform boundedness, permanence and existence of at least one positive periodic solution together with it's global asymptotic stability. The existence of periodic solution has been obtained by using the continuation theorem of coincidence degree theory proposed by Gaines and Mawhin. After establishing the periodicity, we prove the global asymptotic stability of the positive periodic solution by constructing a suitable Lyapunov functional. After analyzing the deterministic model, we formulate the corresponding stochastic model using the perturbation in the growth rate parameters by white noise terms. Then using the existence of positive global solution, we prove that all the higher order moments of the solution to the stochastic system is uniformly bounded which ensure that the solution of the stochastic system is stochastically bounded. We have found easily verifiable sufficient conditions for non-persistence in mean, extinction and stochastic permanence of the stochastic system using Itô's formula. Sufficient condition for permanence shows that if the noise intensity is very low then the solution of the stochastic system persists in the periodic coexistence domain of the deterministic system. But the sufficient conditions for extinction shows that the noise of high intensity can break the periodic coexistence of the solution to the deterministic system and drive the system to extinction. At the end we perform exhaustive numerical simulations to validate our analytical findings.