MATHEMATICAL ANALYSIS OF PREY PREDATOR MODELS WITH HOLLING TYPE I FUNCTIONAL RESPONSES AND TIME DELAY

. We examine two prey and one predator models with Holling type I functional behaviours in this paper. To demonstrate the system’s permanence and boundedness, we used a discrete-time delay. Through the use of traditional mathematical techniques, the effects of random variations in the environment and time delay on the model’s stability are analytically examined. The stability and Hopf-Bifurcation for the competition model are also described and shown. A few numerical computations are provided to demonstrate the efﬁcacy of the theoretical ﬁndings.


INTRODUCTION
Mathematics plays a major role in biology with the help of a biological models [1] - [2]. All areas of ecology have seen a significant increase in mathematical developments in population biology, which have a long history of being created by mathematics research into the dynamical characteristics of population developments. Based on the existence and significance of predators and prey in nature, numerous authors have created mathematical models of the relationship between the two [3], [4]. The predator-prey interaction model is the main focus of this work.
Predator-prey competition is based on interactions between two species and how they affect one another [5]. During the prey predator competition, there are various types of interactions between the species. Numerous mathematical models have been constructed to represent the dynamics of prey-predator systems as a result of substantial research. The functional response, which describes how the predator's feeding rate changes with regard to the prey density, is a crucial component of these models. The Holling Type I functional response is a prevalent and well-known type of functional response among the several functional response types. In our work, we study Holling type I functional response [6] to bring two prey and one predator into the conflict. A mathematical model with Holling type I functional response describes the connection between a predator's prey density and consumption rate. Assuming that the predator's consumption rate is directly proportional to the prey density up to a certain saturation point, it is one of the most fundamental functional response models. After this, even if the prey density rises further, the predator's consumption rate stays constant.
In population ecology, dynamics of predator-prey systems is crucial [7]. It establishes how various species are distributed within the environment and, in some cases, forecasts whether a particular species will flourish or go extinct. Time delay, in addition to functional response, has a considerable impact on the dynamics of prey-predator systems. Time lags can occur as a result of a variety of biological and environmental conditions, such as the time it takes the predator to seek for and capture prey after encountering it. These delays inject memory effects into the system, resulting in complicated dynamics that differ markedly from those reported in delay-free models. In order to represent and take into account the necessary reaction time, gestation period, feeding time, etc., delay differential equation DDEs have a long history of modelling prey-predator systems [8] [9], [10]. By considering multiple delays, Kundu and Maitra [11] developed a three-species predator-prey system with cooperation among the prey.
They investigated how time delays affected the system and used time delays as the bifurcation parameters to derive the necessary conditions for the existence of Hopf bifurcation.
As a result, the mathematical analysis of prey-predator models with Holling Type I functional responses and time delay is the subject of this work. We intend to research the effect of time delay on the system's stability and bifurcation behaviour, as well as how it effects the coexistence or extinction of predator and prey populations. We aim to gain insights into the complicated dynamics shown by these models by using mathematical tools such as stability analysis, bifurcation theory, and numerical simulations. Understanding the behaviour of prey-predator systems with Holling Type I functional responses and time delay is of theoretical interest, but it also has practical relevance in ecology and conservation biology. It can help us better understand the repercussions of predator-prey interactions and contribute in the development of effective management and conservation measures. Overall, this study lays the foundation for further research into the mathematical properties and ecological implications of prey-predator models with Holling Type I functional responses and time delay, thereby improving our understanding of the dynamics of complex ecological systems and their conservation.
In this work, we investigate the dynamics of a two-prey one-predator delay differential model with Holling type I functional response. In section 3 and 4, we discuss about the positivity and boundedness of the model. We discuss about stability analysis without delay in section 5. Similary we discuss about stability analysis with delay in section 6. Finally, numerical simulations were performed to determine how the population of the species that competed changed dramatically in section 7.

MATHEMATICAL MODEL
Consider the following model, with initial conditions where u 1 (t), u 2 (t) and v(t) represent the density of prey 1, prey 2 and predator populations. m 1 and m 2 are the intrinsic growth rates of prey 1 and prey 2; The carrying capacities of prey 1 and prey 2 are represented by K 1 and K 2 ; w 1 denote the competition coefficient of prey 2 on 1 and w 2 denote the competition coefficient of prey 1 on 2; λ 1 , λ 2 are rate of predation on prey 1 and prey 2; δ denote the death rate of predator; ζ denotes the predator's decreased rate as a result of intra-specific competition. Throughout this work, the time delay parameter is represented by τ.

POSITIVITY
Theorem 1. For every solution of (1) with initial conditions (2) Proof. Using the initial conditions (2), for t ≥ 0, we have

BOUNDEDNESS
Theorem 2. All the solutions of system (1) with positive initial values are bounded.

STABILITY ANALYSIS WITHOUT DELAY
5.1. Local Stability. The non -linear matrix of (1) which is evaluated at the interior equilibrium point is given by Characteristic equation of (6) is, Here M 1 , M 2 and M 3 are given by By Routh Hurwitz Criterion, the system is locally asymptotically stable, if η 1 > 0, η 3 > 0 and η 1 η 2 − η 3 > 0 are satisfied.

Global Stability.
Consider the Lyapunov function [12] for demonstrating the global asymptotic stable behaviour.

RANDOM FLUCTATION ANALYSIS USING WHITE NOISE
We permit stochastic perturbations of the variables u 1 , u 2 , and v around E * in this section if it is locally asymptotically stable. We consider white noise stochastic perturbations that are proportional to u 1 , u 2 , and v distances from u * 1 , u * 2 , and v * . As a result, the stochastically perturbed system with t is given by where υ i , i = 1, 2, 3 are real constant and κ i t , i = 1, 2, 3 are standard Wiener processes that are independent.
We consider the linear system of (19) around E * in order to conduct the following analysis on E * stochastic stability: where Theorem 3. If there is a function Y ∈ C 0 2 (X) that satisfies the subsequent criteria, for t ≥ 0 the trivial solution of 20 is exponentially p-stable.
The trivial solution of 20 is globally asymptotically stable if p = 2 in (22). The proof is similar to the theorem in [13].
then the zero solution of (20) is asymptotically mean square stable.
Proof. Consider the Lyapunov function For p = 2 inequalities in (22) are true.

NUMERICAL ANALYSIS
By randomly choosing appropriate and suitable sets of parameters, we evaluated the conditions, particularly the stability and impact of white noise, that were carried out in the preceding sections. Here, we use Mathematica to run numerical simulations to validate our analytical results for system (1).

Case 1: Simulation in the absence of delay
Here we considered (1)  Upper panel depics that only the second prey and predator populations are alive, while the first prey population has gone extinct. One can easily identify that the system of equations which is free from the time delay terms is always stable which is shown in Lower panel.

Case 2: Simulation in the presence of delay
Here we considered (1) with time delay. When time delay increased to 0.01, Figure 2 a) shows that only the first prey and predator populations are alive, whereas the second prey population has gone extinct. In Figure 2 b) all three populations coexist simultaneously. Further when time delay is increased to 0.04, a periodic solution occurs between prey 2 and the predator while prey 1 remains at zero level and vanishes.
When τ = 1, in Figure 4a), prey population will become extinct while a stable behaviour exist between prey 2 and the predator. Similarly, the two prey populations exist in Figure 4 (b), but the predator population has vanished. Figure 5 and Figure 6, exhibits a periodic solution between all the three populations.

CONCLUSION
The interaction of two prey and one predator in an ecosystem with a discrete-time delay and a Holling type I functional response has been investigated. We examined the well-posedness of the system, such as positive invariance and boundedness. The stability analysis was conducted both locally and globally, with and without a time delay. Descartes' rule and Buttler's lemma are also used to describe and prove the Hopf -bifurcation characteristics. Finally, numerical simulations were run to determine how the population of the species that competed changed dramatically.