Effect of white noise and Diffusion on the dynamics of predator- prey density dependent model

 

K. Shiva Reddy_1*, C.V. Pavan Kumar_2,  M. N. Srinivas_3, M. A. S. Srinivas₄

¹Department of Mathematics, Anurag Group of Institutions, Venkatapur, Hyderabad-500 088, India

²Department of Mathematics, Vignan  Institute of Technology  and  Science, Vignan Hills , Hyderabad , India

³School of Advanced Sciences, VIT University, Vellore-632 014, India

⁴Department of Mathematics, Jawaharlal Nehru Technological University, Kukatpally, Hyderabad.

 

ABSTRACT:

In this paper, we consider an ecological model consisting of prey and predator along with special effect of diffusion and random perturbations. The Routh-Hurwitz principle and Lyapunov’s function are employed to analyze the local and global stability of the system respectively, without delay and diffusion, around the positive equilibrium point. The stability of the system is analyzed with delay and without diffusion. Diffusive instability is also verified by perturbation technique and Routh-Hurwitz criteria. The analytical estimates of the mean-square fluctuation of population have also been calculated to explore the dynamics of the system in the presence of Gaussian additive white noise. Computer simulations through MATLAB have been carried out to illustrate the analytical results with suitable numerical examples.

 

 

 

1. INTRODUCTION:

In general, prey- predator models are essential structural slabs that describe the dynamics of ecological systems. In nature, very often species compete among themselves for the purpose of looking for resources to gain an upper hand and hold their own struggle for existence. Their destruction is often the result of their failure or inability to procure the minimum level of resources essential for species survival. Prey predator models can take the forms of resource-consumer, plant-

 

herbivore, parasite-host etc. depending on their intrinsic growth rates. The oldest prey predator model was proposed by Alfred Lotka [1] in 1925 on the strength of various assumptions such as (i) density independent exponential growth of prey species in the absence of predators, (ii) constant per capita mortality rate of predators, (iii) constant conversion rate of predators and a constant predation rate. The prey population is not self- limiting, this model does not explain the real behaviour observed in the environment. Later, in 1930, Volterra [2] revised the model with the inclusion of self- interaction term, which employs logistic growth instead of exponential growth.

 

In 1963, Rosenzweig and MacArthur improved the Lotka-Volterra model by  including the assumption that the predation rate is no longer presumed to be proportional to prey density.

 

There are many realistic situations in environment which prove that the predator species control the number of prey species. Because of high local density of prey, the predators are frequently facing the problem of changing their consumption rates. This is nothing but a type of relation between rates of food consumption of a predator to the density of prey. This is also known as functional response, a term introduced by Holling. He investigated the act of predation and categorised it into the major components like search, capture, handling and digestion. Many authors [3, 4, 5, 6, 7] suggest that the functional response of a species may take one of four forms called the Types I, II, III and IV functional responses. The prey consumption that rises linearly with prey density to a threshold level is type-I. In type II functional response, the attack rate increases at a decreasing rate with prey density until it becomes constant. Prey consumption remains low until a threshold density is reached in the case of Type III response, that is, the predation rate increases exponentially until it levels out. The predator disregards the prey when densities are very low in order to make chasing energetically profitable. When the nutrient concentration reaches a high level, an inhibitory effect on the specific growth rate may occur which is type IV category.

 

In the past three decades, the prey-predator schemes play a significant role in the modeling of population dynamics [8, 9]. Several models of population growth were studied with time delays [10–15]. Several authors [16–21] studied stage-structured models with various attributes like time delay, stochastic, diffusion etc. In many ecological systems, the species may disperse spatially as well as evolving in time. The diffusion arises from the tendency of some species to migrate towards other regions having low population density because of the limitation of resources. Recently, great attention has been paid to the effect of scattering of population in a bounded environment. Many authors [22-30] directed their attention to the diffusion–reaction systems.

 

The Rosenzweig-MacArthur model uses the Holling II type functional response [31] which belongs to the category of models with handling times of predator and in which the predation rate is linearly proportional to prey density. Many authors [32, 33] investigated the dynamics of this model in various directions and this is what inspired us to propose a delayed diffusion and stochastic extension, which are analysed in the consequent sections.

 

 

 

9. CONCLUDING REMARKS:

In this work, we consider an ecological prey - predator model with spatial effect of diffusion and random perturbations. We studied the stability of diffusive delayed model around the positive equilibrium point. The stability of the system is analyzed with delay and without diffusion. Diffusive - instability is also verified by Routh-Hurwitz criteria. We also calculated the analytical estimates of the mean-square fluctuation of population to explore the dynamics of the system in presence of Gaussian additive white noise. Computer simulations through MATLAB figures (1(a)-6(b)) have been done to illustrate the analytical results with the suitable numerical attributes.

 

10. ACKNOWLEDGMENT:

This research is carried out under the UGC Minor research Project (MRP-4617/14(SERO/UGC)), Govt. of India. The Authors are thankful to UGC for financial support.

 

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Accepted on 22.07.2016        © RJPT All right reserved

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