Abstract
We have discussed the dynamical behaviour of a single-species population model in a polluted environment which describes the effect of toxicants on ecological system. Boundedness, positivity, and stability analysis of the model at various equilibrium points is discussed thoroughly. We have also studied the effect of single discrete delay as well as double discrete delays on the population model. Existence conditions of the Hopf bifurcation for single time delay are investigated. The length of delay preserving the stability is also estimated. The direction and the stability criteria of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. The stability of the model with double time delays is investigated by using the Nyquist criteria. By choosing one of the delays as a bifurcation parameter, the model is found to undergo a Hopf bifurcation. Some numerical simulations for justifying the theoretical results are also illustrated by using MATLAB, which shows the reliability of our model from the practical point of view.
Highlights
In the world today, the pollution of the environment is a threatening problem due to the rapid development of industrialization
Samanta and Maiti [9] studied a dynamical model of a single-species system in a polluted environment under two cases: constant exogenous input of toxicant and rapidly fluctuating random exogenous input of toxicant into the environment by means of ordinary and stochastic differential equations
When γ = 1.25 using the parameter values given in Table 2, it is observed that the equilibrium E2 = (0, 4.28082, 12.5) exists and is locally asymptotically stable with hpq < γ, but the interior equilibrium E∗ does not exist, which is shown
Summary
The pollution of the environment is a threatening problem due to the rapid development of industrialization. The study of deterministic dynamic population models with toxicant effect was proposed by Hallam and his colleagues in 1980s [1,2,3]. This model was revisited by many researchers [4, 5]. We have developed a single-species population model in polluted environment. We have discussed the existence and stability analysis of various equilibrium points under zero-exogenous input and nonzero constant exogenous input. The analysis of double delayed model is discussed.
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