Effects of Stochastic Analysis on Development and Maturation of an Ecological System

Communalism, neutrality, predation, and mutualism are all part of an ecological system, as are interactions between organisms, such as harvesting prey and killing predators, and changes in the environment. The primary goal of this chapter is to determine the system dynamics by incorporating random sounds into the model under the assumption of stochastically varying driving forces for species evolution. The existence of fluctuating driving forces in a traditional ecosystem assumes to influence the species' growth: 1,2,3,4 I s I ± time 't' The figure (3.1) is the device where four species are living together with the following assumptions: (i) The system comprises of a prey (S1), predator (S2), two hosts S3 and S4 (ii) S1 is prey of S2 and S1 is neutral to S3 (iii) S2 is predator of S1 and S2 is commensal to S3 (iv) S3 is host of S2 and S3 is mutual to S4 (v) S4 is mutual to S3 and S4 is neutral to S2 (vi) Harvesting of S1 and S2 results in the following stochastic system with ‗additive noise‘. In conjunction with its stochastic analysis, the local and global stability of the deterministic model were applied around the coexisting State. The system also derives bionomic harmony. In addition, the results are validated using MATLAB numerical simulations.

Let be the population densities of species respectively at time instant ‗ t ‘. Let be the natural growth rates of species respectively. Keeping these in view and following, the dynamics of the stochastic system may be governed by the following nonlinear differential equations:

Where are self-inhibition coefficients of species respectively a12 is the interaction coefficient of S1 due to S2, a21 is the interaction coefficient of S2 due to S1 , a23 is coefficient of commensal for S2 due to the host S3 , a34 is the rate of increase of 3 S due to the interaction with S4, a43 is the rate of increase of S4 due to the interaction with S3, K1 , K2 , K3 and K4 are the carrying capacities of species respectively, where are the catch ability coefficients of species respectively. E1 , E2 are the efforts applied to harvest the species S1, S2 respectively. and is a four dimensional Gaussian white noise process and  are Kronecker and Dirac delta functions respectively. In addition to the variables x , y , z , w the model parameters are assumed to be non-negative constants.

In the absence of randomly fluctuating driving forces on the growth of the species, the model system reduces to

The community matrix of the system is At the interior equilibrium point Now using the Routh- Hurwitz criteria the following theorem is stated: Theorem 1: The interior equilibrium point is locally asymptotically stable when

Theorem 1: The interior equilibrium point is globally asymptotically stable if Proof: To find the conditions for global stability at the Lyapunov function is constructed: The system is globally stable if the derivative of Lyapunav‘s function V is negative definite, that is if the matrix A is positive definite, that is if the principal minors of A (say) , 1,2,3,4 M i i  are positive. The principle minors are positive if Hence, the system is globally stable in the above parametric space.

It is the combination of biological balance and economic balance. In the biological balance is given by If the net benefit from the sale of the yielded biomass is equivalent to the total cost, then the bionomic equilibrium is reached. The constant harvest cost of species S1 by unit effort should be c1, and of species S2 by unit effort, constant harvest cost. Let p1 be the price per unit biomass for the species S1 and p2 is the price per unit biomass for the same species S2. The revenue at any time is given by Now if then the economic rent obtained from the system becomes negative and the system will be closed. Hence for the existence of bionomic equilibrium, it is assumed that

A natural phenomenon is hard to explain as a deterministic model, and particularly the aquatic ecosystem, which still has the unpredictable environmental fluctuations. The stochastic analysis lets you gain an insight into every ecosystem's dynamics. The deterministic model with random ambient noise effect leads to a stochastic model in which the system parameters oscillate over their mean values. The point of balance that is supposed to be constant therefore now bounces around the average state. The random noise incorporated in the form of additive Gaussian white noise to the model and then any parameter ' ' p of the system reduces to is the amplitude of the noise and is the Gaussian white noise process. This analysis is mainly to depict the dynamics of the system around the equilibrium point; therefore we linearize the model using the following perturbation method: Analytical evaluation of the population variance is difficult, but it can be evaluated numerically for different set of values of parameters.

Mathematics modeling is a powerful tool in the fields of ecology and epidemiology. Appropriate mathematical models can be used to study a wide range of ecological phenomena, such as prey-predator interactions, intraspecific competition, the transmission of infectious diseases, and their subsequent vaccination, treatment, and other controls. Multispecies ecosystems are complex dynamical systems because they feature many different species or components that interact with one another in a wide variety of ways. The dynamic behaviors of several plant, insect, and animal species have contributed greatly to the advancement of dynamic models of complex ecosystems (both deterministic and stochastic). Deterministic dynamic models are used to describe system properties such as stability, instability, bifurcations, and catastrophic state shifts. The study examines the dynamics of change through time, the perseverance of stable states, and the transformation of stable states by parametric change (bifurcation), among other topics. A key concept in ecosystem analysis is that of stability. Many instances of ecological interaction with the word "instability" are possible. As a statistically formalized version of the more general scientific concept of a deterministic process, the idea of the dynamical system is extremely important. The present and the laws governing the evolution of

do not evolve over time, the initial conditions would fully characterize the system's behavior. Because of this, we can define a dynamical system as something that has both a collection of possible states and a law governing the evolution of that state through time. We then define a dynamical system by combining our previous definitions of its constituent parts.

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Research Scholar, Shri Krishna University, Chhatarpur M.P.