DYNAMICS OF PREY PREDATOR MODEL WITH PREY REFUGE USING A THRESHOLD PARAMETER

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. A lot of ecological studies deal with the coexistence of interacting species. The dynamics of this interaction helps in proper understanding of specie coexistence and resource conservation. In this work, we considered a prey-predator model with Holling type II functional response with threshold parameter incorporating a prey refuge. In order to understand the specie coexistence, we analyzed the long term dynamics of the model using a threshold parameter P0 and a prey refuge parameter mε[0,1). We discussed the stability of the steady states and also carried out some numerical simulation in order to validate our analytical results.


INTRODUCTION
Ecologists are often confronted with handling issues like the net effect of predation on the population dynamics of prey, variations in biomass of both prey and predator due to interaction, conditions for coexistence of interacting species using some parameter combinations and so on. The preservation of the ecological niche of species is salient to ecologists alongside the longevity of interacting species. Dawed et. al [2020] recommended that coexistence among predators and prey in the same environment is possible provided a good management of some factors (such as contacts between species, additional food supply, growth rate of species, etc).There are many ecological systems which describe the interaction between predator and prey and proper understanding of how the species coexist is a challenge in ecology (Holling, [1965]; Kar  Moustafa et al, [2018]). There are a lot of research works that have studied the dynamics of ecological systems with concentration on single predator and prey populations (Rosenzweig and MacArthur,[1963]). Kar [2006] looked at modelling and analysis of a harvested prey-predator system incorporating a prey refuge.

PREY-PREDATOR MODEL FOR A COEXISTENCE ECOSYSTEM
We shall look at a single prey-predator system. In this work, we note that the ecological niche of the two species are not affected by external factors such as drought, fires, epidemics and so on. Besides the model description assumption and definition of variable and parameters, it is assumed that a constant proportion m ∈ [0, 1) of the prey can take refuge to avoid predation.
This leaves (1 − m)x of the prey available for predation and hence the model under the above assumptions with Holling type II functional response is given by: A real life example is the interaction between the zebra (x) and the lion (y). The description of the variables and parameters that are used in the model (1) can be seen in Table 1. (1). We obtain the following equilibrium points from the system (1) as follows:

Variables/Parameters
Meaning Unit  (1) We remark that for each of the equilibrium points E i (for i = 1, 2, 3) to exist, the inequalities 0 ≤ x ≤ k must be satisfied by each of them. Furthermore, for the equilibrium point E 3 to exist, it must each satisfy the inequality: We make use of a threshold quantity (predation number denoted by P 0 ) which gives a condition for the stability of the system around the equilibrium points. This quantity can also be likened to the basic reproduction number R 0 (Van Den Driessche and Watmough, [2002]) in epidemiological models. Its calculation is done in like manner by the approach of the next generation matrix (Collins and Duffy, [2016]): In ecological terms, we can describe P 0 as the combination of parameters which quantifies the prey consumption required for survival. So, P 0 = 1 implies that the predator uses prey biomass at a rate almost corresponding to their own biomass loss. For P 0 < 1 less prey is ingested per unit of predator biomass loss. For P 0 > 1 more prey is devoured per unit of predator biomass loss. (1). The investigation of the stability of this model is a tool that will assist us in describing the entire dynamics of the system. We will examine the stability using the threshold quantity P 0 .

Stability analyses of model
Theorem 1. The equilibrium point E 1 is unstable no matter the value of P 0 .
Proof. It suffices to show that at least one of the eigenvalues of the Jacobian of the model Proof. When we evaluate the eigenvalues of (1) at E 2 we obtain Similarly, in terms of the prey refuge parameter, the system will be stable if Theorem 3.The equilibrium point E 3 is stable if P 0 > 1.
Proof. We shall make use of the characteristic equation. This is given as: 1+a(1−m)x * = A, then we will have that The coexistence equilibrium point is stable if λ 1 + λ 2 is negative and λ 1 λ 2 is positive.
Similarly, using the product of roots: We then have

MODEL SIMULATION AND DISCUSSION OF RESULTS
In order to validate our analytical results, we examine the long-term dynamics of the model by carrying out numerical simulations using the parameter values given in Table 2.
Parameters Parameter values Source    Table 2 and varying γ (Figures 1-2) and m (Figures 3-5). γ is varied because its alteration easily affects the predation number. For Figure 5, γ = 0.45 results in P 0 = 0.889. For Figure ??, γ = 0.3 results in P 0 = 1.333. From Figure 5, the trajectory of the predator biomass increases and then decreases due as the prey biomass decreases. So, less prey is devoured per unit of predator biomass loss. From Figure ??, when the predation number is greater than one, the When we understand the dynamics of certain ecosystem, it helps in the preservation of the coexisting species. An insight into a homogeneous community of a single predator and prey helps to study the dynamics of their interaction and useful in analysis. We illustrated this using x as the prey with y as the associated predator. In order to identify system stability, we computed a threshold parameter, P 0 (Predation number) which summarizes most of the parameters. Also, we see that the prey refuge parameter also contributes to investigating the survival of both prey and predator. A possibility to introduce heterogeneity in the population will change the entire dynamics and generate more interesting result for specie coexistence. In general, this work can be extended to a heterogeneous population in order to account for multiple species coexistence.
We can also use a different functional response apart from Holling type II which will give us a widespread view in our analysis.

Authors' Contribution.
This present work discussed the coexistence of both species using both a threshold parameter and a prey refuge parameter. A threshold quantity, the predation number P 0 was introduced to evaluate prey consumption per equivalent of predator biomass (Collins and Duffy,[2016]). In addition to this, we investigated how to promote long term coexistence of both species in relation to the prey refuge parameter mε[0, 1). The first and corresponding author originated the work, supervised and performed the simulations. The co-author contributed also by handling the analysis.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.