**Project Draft**

# Effect of Seasonal Growth On Delayed Prey-Predator Model

**Article**

This articles subject is a review of the paper Effect of seasonal growth on delayed Prey predator model
Authors: Sunita Gakkhar, Saroj Kumar Sahani, Kuldeep Negi. We are not taking the lemmas and theorems also represent simulation in one way

## Background

The typical prey-predator model is always a main interest of mathematicians and ecologists due to its universal importance and existance. The Lotka-Volterra model describes interactions between two species in an ecosystem, a predator and a prey. This represents first multi-species model. Since we are considering two species, the model will involve two equations. Factors that are necessary for time delay are age, death rate, birth rate, feeding time, reaction time, food storage, and hunger coefficients in predator-prey interactions. In this paper the author's main focus is for the effect of seasonality in delayed prey-predator system and the effect of time delays due to negative feedback of prey species to the growth of species itself. (Delayed Predator-Prey model also discussed in MBW:Predator-Prey Models With Delay And Prey Harvesting[1]

## MODEL

Here we describe the mathematical model, parameters, and initial conditions.
model of prey predater system .the delayed occure in growth as well as in interaction .
for this it is assume that the pray takes time t1 to convert the food in to the growths [11],where as the predater takes time t2 for tje same [8,9 15].accordingly ,the delay-predater model can be expressed as
the state variable u1(t1) and u2(t2) denote the density of the prey and predater species, respectively .all the parameter in the model take positive value ,i e a0>0,d>0,d1>o .

**local stability analysis**
the system has two boundary equilibrium points R(0,0) and R2(1,0).
700px

## Result

**numerical simulation**

In this section, we discuss a picture in the article and present our version of the picture.

**parameter**

a0 = 5.0;

a = 0.5;

d = 0.4;

d1 = 2.0;

The pictures given below for the different value of delayed .

when delays are T1=0.5,T2=0.5 system showing the stable behavior of the solution.

File:A11.jpg

when delays are T1=0.5,T2=0.8] system showing the stable behavior of the solution.

File:A12.jpg

when delay are T1=0.5,T2=1.5 the system becomes unstable .

File:A13.jpg

when delays are T1=0.7,T2=0.5 system showing the stable behavior of the solution

File:A14.jpg

when delays are T1=1.2,T2=0.5 system showing the stable behaviour of the solution

File:A15.jpg

when delays are T1=1.8,T2=0.5 the system becomes unstable .

File:A16.jpg

**Effects of seasonal growth of prey**

When the environmental factors that affect various parameters of the ecological model fluctuate periodically, then the corresponding parameters should be taken as periodic functions of time. so for the effects of seasonally by superimposing a periodic growth rate on the original intrinsic growth rate of prey as 1 + a1 sin (ωt). The constant a1 is the “degree” of seasonality and ω as the angular frequency of the fluctuations caused by seasonality. system is now change to following form

700px

For numerical simulation, consider the set of parameters as

a0 = 5.0;

a = 0.5;

a1 = 0.8;

d = 0.4;

d1 = 2.0;

The bifurcation diagrams with respect to ω are drawn in following for the following three cases: (a) τ1 = 0 and τ2 = 1.5 (b) τ1 = 2.0 and τ2 = 0 and (c) τ1 = 2.0 and τ2 = 1.5. In all the three cases the dynamical behavior is observed with periodic variation of parameters in the delay models.

File:Delay2.jpg.

**Fig1**

For these values of parameters the bifurcation diagrams with respect to a1 are drawn in below for the three cases (a) τ1 = 0 and τ2 = 1.5 (b) τ1 = 1.4 and τ2 = 0 and (c) τ1 = 1.4 and τ2 = 1.5. The chaotic behavior of the solution increases with increase of a1 in all the three cases as

File:Delay1.jpg.

** Fig2**

keeping ω = 2.0, the bifurcation diagrams with respect to parameter τ1 in the range 0 τ1 3 for different values of τ2 as (a) τ2 = 0.0, (b) τ2 = 0.5, (c) τ2 = 1.0 and (d) τ2 = 1.5 are shown in Figs3 a–d. It clearly shows increase in the chaotic behavior of the solution with increase of τ1 as well as τ2. In Figs. 3a and b, a wide 1-periodic window is observed and in Fig. 3c two wide 1-periodic windows are clearly visible. But for higher value of τ2, only small sizes of windows are shown in Fig. 3d. Further, in these figure the size of 1 periodic window is decreasing with increasing τ2 and Bifurcation point is progressively shifting towards left.

700px.

** Fig3**

Consider ω = 2.0 and the other parameters are same as before, the bifurcation diagrams with respect to parameter τ2 in the range 0 τ2 3 for different values of τ1 as (a) τ1 = 0.0, (b) τ1 = 0.5, (c) τ1 = 1.0 and (d) τ1 = 1.5 are shown in Figs. 4a–d. Here also the bifurcation point shifts towards left with increasing τ1. Wide periodic window are obtained for τ1 = 1.0 and but for higher value τ1 = 1.5 these periodic windows are vanished and the solution becomes chaotic.

700px.

** Fig4**

**== Result ==**

A delayed prey–predator model has been analyzed both analytically and numerically. The studies have shown that when positive equilibrium point exists then the predator free equilibrium point is always a saddle point.
if d > 1/(a + 1).
when the death rate of predator increases then the solution becomes a stable periodic solution

## Matlab Code

In this section, we provide the matlab code to regenerate the above picture. u1u2d.m