# Single Species Population Models

Global stability single species population models for nonzero exogenous inputparameter values are given in Table 1. Discussions and Conclusions In this paper, we have developed a single-species population tanzkurse fur singles offenbach under the influence of kostenlos frauen kennenlernen in berlin in the population and in the environment by means of ordinary differential equations in terms of their concentrations.

It is shown that the model system 3 with 4 is bounded and the solutions are positive, which in turn implies that the system is ecologically well behaved. The analysis of the existence of various equilibrium points under zero exogenous toxicant input and nonzero exogenous toxicant input leads us to Theorems 2 and 3respectively. Theorem 2 states that under zero exogenous input the trivial equilibrium is unstable, the interior equilibrium is not feasible, and only the axial equilibrium is locally asymptotically stable.

On the other hand, when exogenous input is nonzero, the existence of the interior equilibrium depends on some conditions and the equilibrium is locally asymptotically stable under some condition. The stability criteria for given in Theorems 4 and 5 are the conditions for locally and globally stable coexistence of the population biomass, population toxicant, single species population models environmental toxicant. It is proved by several researchers that the effect of time delay must be taken into account in the population models to make them ecologically more meaningful and useful.

It seems also reasonable that the single species population models of the environmental toxicant on the organismal activities and the population growth will not be instantaneous. It must be mediated with some discrete time lag required for incubation. From this point of view, we have formulated single and double time delayed model given in 31 and a and brespectively, where the delays may be looked upon as the reaction time of the environment toxicant on the population biomass and the population toxicant.

The analysis of the delayed models 31 and a and b shows more complicated behaviour than their nondelayed counterpart. The rigorous analysis of the single delayed model 31 leads us to Theorem 6 which mentions that the stability criteria in absence of delay are no longer enough single species population models guarantee the stability in the presence of delay, but rather there is a critical value of the gestation delay such that the system is stable for and becomes unstable for at the interior equilibrium.

From *single species population models* next part of our analysis, we obtain the estimated length of delay to preserve stability at and the direction of Hopf madchen kennenlernen uni by using normal form theory and center manifold theorem. The stability of the double delayed model a and b is investigated by the Nyquist criteria single species population models leads us to Theorem 9. By choosing one of the delays as a bifurcation parameter, the afrikanischer mann sucht deutsche frau a and b is found to undergo a Hopf bifurcation under some conditions stated in Theorem Analytical studies can never be completed without numerical verification of the results.

So, some numerical simulations for justifying the important theoretical results are also illustrated by using MATLAB, which shows the mathematical and ecological reliability of the proposed model. Ma, The Study of Biology Model, Anhui Education Press, He fei, China, View at Google Scholar Z. View at Google Scholar B. Di Liddo, and I. De Luna and T. An International Multidisciplinary Journal, vol.

Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, The Netherlands, View at MathSciNet Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, **Single species population models** Press, New York, NY, USA,

## Sustainability in single-species population models

### Journal of Mathematics

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