^{1}

^{*}

^{1}

^{*}

The current study investigates the predator-prey problem with assumptions that interaction of predation has a little or no effect on prey population growth and the prey’s grow rate is time dependent. The prey is assumed to follow the Gompertz growth model and the respective predator growth function is constructed by solving ordinary differential equations. The results show that the predator population model is found to be a function of the well known exponential integral function. The solution is also given in Taylor’s series. Simulation study shows that the predator population size eventually converges either to a finite positive limit or zero or diverges to positive infinity. Under certain conditions, the predator population converges to the asymptotic limit of the prey model. More results are included in the paper.

The predator-prey problem has been interesting to many researchers [

Predation can increase, decrease, or have little effect on the strength, impact or importance of interspecific competition [

It is discussed in [

The predator-prey problem with the assumptions of little or no effect of predation on the prey population growth is studied in [

There are several options to consider among the generalized growth models [

The following sections are presented as follows: predator-prey models are presented in Section 2; Gompertz model in Section 3; solution for the Predator-prey equations in Section 4; simulation study in Section 5; analysis of phase diagram and equilibrium points in Section 6; and conclusions in Section 7.

The classical Lotka-Volterra predator-prey model is given by:

where

In the present work, we consider the case when the interaction of the prey and predator populations leads to a little or no effect on growth of the prey population, that is

where

The prey Equation in (2) is the first order differential equation. The solutions of this first order differential equation are studied as growth models in [

The general approach for solving Equation (2) consists of the following steps:

1) Assume that the impact of predator on prey population growth is negligible,

2) Predator population declines in absence of prey,

3) Predator population grows with a rate proportional to a function of both

4) Assume that there is prior information about the prey population that

5) Solve for predator population growth

We assume that the growth of prey population follows Gompertz growth model and construct the corresponding predator growth function. The Gompertz model is given in [

where

Here, we solve the ordinary differential equations in (2), then determine intersection points at which the prey and predator population attain same values, and finally three special cases of predator population are considered.

The solution for the ordinary differential equation in (2) is derived assuming that prey follows the Gompertz model in (3). After substituting (3) in (2), the corresponding predator population growth function is derived to be:

Or equivalently,

where

Equation (4) is also equivalent to the following solution (6)―that can be expressed in terms of the well known exponential integral function Ei:

Note that the exponential integral function is a popular function that is often useful in many applications. We believe that this respective predator population growth function

The predator models in Equations (4-7) appear to be new functions and they do not match with any one of the commonly known growth models.

Points of intersection are the point of time at which the prey and predator populations attain the same sizes.

Whenever it occurs, let the point of intersection be represented by

Equations (3) and (4). In trying to solve these equations, we get the following expression expressed explicitly as:

where

Equation (7) can only be computed numerically.

To further understand the model in Equation (4), three special cases are identified which are dependent of birth and death parameters of predator population. The cases are considered here below.

Case I

the form

population converges to lower or upper asymptote depending on the initial value of the predator population. The initial population size can be larger or smaller than

It is interesting to note that both the prey and predator population sizes converge to the same asymptote

Case II

Case III

The minimal point at which the predator growth curve turns or gets minimum value is found to be:

The cases can be generalized to a statement that ratio of deaths to births

meter

The simulation study is carried out based on Equation (4). The study is designed by varying the model parameters:

Prey model: Gompertz growth model

Prey model parameters:

Predator model parameters:

Cases: Case I:

Case I:

Case II:

Case III:

The results of the study are displayed in Figures 1-3.

We have shown by simulation study that the predator population either converges to a finite limit or converges to zero or diverges to infinity on the positive side depending on the parameter values under the assumption that prey follows Gompertz growth model. Moreover, for a particular value of birth parameter , the population sizes of both prey and predator converge to same asymptote. These findings are similar with those in [

The newly proposed predator-prey model (2) in its full form can be expressed, in case of Gompertz growth of prey population, as the system of equations

Nature of the Equilibrium Point

The Jacobean matrix at this equilibrium point takes the form

Condition I

Condition II

Nature of the Equilibrium Point

Equilibrium point | Eigenvalue | Condition | Sign of eigenvalue | Nature of equilibrium point |
---|---|---|---|---|

Both | Stable | |||

Unstable | ||||

Stable | ||||

Both | Stable | |||

Unstable |

The Jacobean matrix at this equilibrium point takes the form

Condition I

Condition II

Condition III

Some mathematical aspect of the well known predator-prey problem is studied by modifying the respective classical assumptions. We assume that the prey population growths naturally with no interaction effect due to predation and rate of growth is non-constant. Then, the predator-prey equations are solved considering prey grows as Gompertz model. The solution for the predator population is found to involve the exponential integral function and is equivalently expressed in terms of Taylor’s series.

The simulation studies and further analysis of the models reveal that the predator population grows in such a way that either converges to a finite limit or zero or diverges to positive infinity. There is a situation at which both prey and predator populations converge to the same limit irrespective of their initial population sizes. There is also a situation where the predator population attains a minimal point before it diverges to infinity. Moreover, two equilibrium points are identified which are stable only under some specific conditions.

Derivation of Predator Population Model given Prey follows Gompertz Growth Model

Assume the prey population growth can be represented by the Gompertz function:

Then the predator equation can be solved as:

Substituting (i) in (ii) gives

We now introduce a new variable

So as to get:

To evaluate the integral, we now use Taylor’s series expansion of

Hence

Using (iv) in (iii), we get:

To determine the integral constant

To eliminate

But

Thus, (vi) can be rearranged as:

Thus

The relationship (viii) is a phase path equation. It can be used to analyze phase path diagrams.

Show the Solution with Exponential Integral Function and the Taylor’s Series of the Predator Population are Equivalent.

Exponential integral function

Here

On subtracting (iii) from (ii), we get

But the expression

Hence, using (v) in (iv) we get

Using (v) in (i), we get

Or equivalently

Thus (vii) is the required predator equation expressed using exponential integral function

Note that the indefinite integral