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Two extensions of stochastic logistic model for fish growth have been examined. The basic features of a logistic growth rate are deeply influenced by the carrying capacity of the system and the changes are periodical with time. Introduction of a new parameter

Resources generated through bio-reproduction processes and through bio-chemical processes constitute a rare and unique gift of the nature to the human race. These resources are renewable by the very nature of biological processes. Fishery is a prime example of renewable resources that the human race has been exploiting for its survival and shelter since the time immemorial. Fresh water resources are repeatedly renewed through the nature’s recurrent activities, and in turn they renew our agricultural produces and hydro-electric power generation [1-2]. The problem of management is complicated by the factor that the natural populations have a tendency to fluctuate in response to stochastic perturbations in their physical and/or biological environment [

As a matter of fact, after the success achieved by Pearl [

or the corresponding differential equation

where

In the first stage of the extensions, we shall remain confined to the deterministic versions. A close scrutiny of Equation (2) shows that it may be immediately modified in the following ways: Faris Laham et al. [

If

The units of

Further, it has been observed that some fish species are grow rapidly and some are very slowly. Thus, in the former case, the growth curve lies to the left and above of the logistic curve, while in the later case, the growth curve lies far to the right and below the logistic curve. Bearing these points in mind, we propose the second extension of Equation (2) in the form

where

Case-I: Substituting Equation (3) into Equation (2), we obtain

Setting,

Further, assuming that

In order to convert the non-linear Equation (8) into linear equation we have to use the transformation

or

with initial condition

The integration of the non-homogeneous linear Equation (10) and using the initial condition Equation (11), we obtain

where

The asymptotic behavior of Equation (12) will be governed by

Case-II: Regarding the second extension given by Equation (5), we have already pointed out the expected qualitative changes. First of all we observe that Equation (5) is also a non-linear equation of Bernoulli type, hence it can be reduced to a linear equation by setting

Rearrangement of Equation (5) leads to

or

Combining Equations (14) and (15), we find

with initial condition,

The direct integration of Equation (16) gives

or

For large

and

As in the Case-I, we observe here that, in long-run the fishery process effectively forget its initial stage, however, the choice of

Stochastic Formulation of the Extension Model (Case-I): Several stochastic versions of logistic model have been discussed in literature on population processes and in ecology [8-11], and only the steady-state studies have been made. In our problem, we shall first obtain a time dependent solution of stochastic version of the logistic version of the logistic model for fish growth with multiplicative fluctuations. The stochastic differential equation corresponding to Equation (2)

with initial condition

where

where

On setting

with initial condition

Using the concept of White noise, we obtain the drift and diffusion coefficients

The corresponding probability density function

Further, if we transform

or

or

Therefore, on substituting Equation (25a) to Equation (25c) into Equation (24), we obtain

where

For the sake of compactness, we shall rewrite Equation (26) as

where

Therefore, Equation (27) becomes

In this setting, following Wang and Uhlenbeck [

as the probability flux.

Further, following Feller [

Equation (28) with boundary conditions, Equation (30), can be solved either by using the method of separation of variables or by applying the Laplace transformation technique. In the former case the partial differential Equation (28) is transformed into two ordinary differential equations of order one and two, whereas in the later case we obtain a single non-homogeneous ordinary differential equation of order two. However, the second approach becomes tedious and involved for two reasons. Firstly, to solve the Laplace transform of Equation (28), we have to construct suitable Green’s functions; and secondly the Laplace inversion of the solution so obtained in itself is a formidable task. In our study, therefore, we shall adhere to the former method.

We split up our probability density function

where

As we have considered

and therefore Equation (32) reduces to

or

In Equation (34), the left hand side is a function of

and

Now we have two ordinary differential equations Equation (35) and Equation (36) to replace Equation (27). Further, we observe that Equation (27) represents an eigenvalue problem. The boundary conditions Equation (30) imply

or

Using Equation (33) in Equation (37), we obtain the required boundary conditions

Since

where

Substituting

Whence

and

We can easily show that, treating

With an appropriate identification of parameters, Equation (40) turns out to be an Erlang distribution and, thus in the steady-state the mean and variance can be directly evaluated. Thus on setting

Therefore,

and

In this paper, first we have investigated the logistic growth rate when it is influenced by the carrying capacity of the system and have analyzed the modified logistic model for fish growth. It is to be highlighted that the different stochastic versions of the logistic model and its extensions can be extended further in several directions. One may examine the threshold effect through logistic model and one can also examine the effect of stochasticity through the parameters

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