Existence of Solutions for Front-Type Forced Waves of Leslie-Gower Predator-Prey Model in Shifting Habitats

Existence of Solutions for Front-Type Forced Waves of Leslie-Gower Predator-Prey Model in Shifting Habitats

Theint Theint Thu

Department of Engineering Mathematics Polytechnic University (PU Myitkyina), Myitkyina, Myanmar

Ei Ei Kyaw

Department of Engineering Mathematics Polytechnic University (PU Myitkyina) Myitkyina, Myanmar

Abstract – In this paper, the definitions of front type and a

pair of generalized upper and lower solutions are firstly described. Then, Leslie-Gower predator-prey model and the

properties of kernel and growth rate function are expressed. The

suitable upper and lower solutions combined with the Schauders *

fixed-point theorem, fatous lemma is utilized to solve the existence of nonnegative solution. Additionally, under the

appropriate parameter assumptions, the existence of front type

forced wave is verified.

Keywords – Schauders fixed-point theorem; fatous lemma; upper and lower solutions; compact; precompact; integration

for all z

\ E for some finite subset E of .

1. INTRODUCTION

   Wave propagation is an important phenomenon in many scientific areas such as physics, biology and ecology. Reaction- diffusion models are commonly used to describe how a quantity, such as a biological population or chemical substance, spreads in space over time. In classical theory, traveling waves

   move with a constant speed that is determined only by the

   III. LISLIE-GOWER PREDATOR-PREY MODEL

   In this section, the forced waves of Leslie-Gower predator- prey model in shifting habitats with nonlocal dispersal is focused. First, we consider the following diffusive predator- prey model with one prey and one predator:

   internal properties of the system. However, in real world

   situation, wave propagation is often influenced by the external

   effects. These effects may include environmental forcing,

   boundary conditions. Front-type forced waves describe wave solution that appear as moving fronts connecting two different

   where the unknown functions u denote the population density of the prey and v denote the population density of the predator

   at position x and time t. All parameters d* , d* , r, h, k are

   stable states under the influence of forcing. 1 2

   positive. Parameters d* , d* represent diffusion rates for prey

   1 2

2. PRELIMINARIES

We begin our article by giving the definition of front- type,

and predator, the function r, represents the growth rate, k

denote the per capita capturing rate of the prey by a predator

a pair of generalized upper and lower solutions.

per unit of time, and v

u h

represents Leslie-Gower terms,

1. Definition

   First, For a scalar wave profile

   , it is called a front type.

   x st, if

   which means that the carrying capacity of the predator is proportional to the population size of the prey.

   The parameters h and k satisfy the conditions

2. Definition

The continuous functions 1 , 2 and , are

1 2 A. Properties of Kernel and Growth Rate Function

called a pair of generalized upper and lower solutions if

We always assume that the kernel functions

inequalities

are bounded functions and satisfy the following

satisfy the following properties:

B. Existence of Nonnegative Solution

and

for

In this section, we state the theorems for the existence of nonnegative solution.

1 Theorem

The growth rate function r()

satisfies the following two

Suppose that s 0 . If 1 , 2

and ,

are a pair of

properties:

(H) r()

is continuous in ,

lim r(z)

exists

upper and lower solutions of (4) satisfying

in ,

satisfying

and r(z) r( )

for all z

. Without

then (4) admits a solution 1 , 2

such that

(H )

loss of generality (up to a rescaling), we choose r( ) 1 ;

there exists C 0 and 0 such that

Proof:

Let X B C ,

for all z , i 1, 2.

be the space of all uniformly continuous

lim

and bounded functions defined in . Then, X is a Banach space equipped with the sup-norm.

We are interested in the propagation phenomena for system (2). We study the special C1 solution of form

we consider the nonlinear operators

where parameter s being the shifting speed of the climatic condition, which is called the forced wave.

Let z st x, and the corresponding wave profile system

to system (2) is as follows

From assumption (H) , the environment is favourable to the prey ahead of the climate change and then gradually

deteriorates until it becomes hostile to the species. This is equivalent to the boundary condition 1,2 0,0.

We shall consider the constant unique coexistence state of

We define the following operator

system (2) such as

We define

Thus, a fixed point of P is a solution of (4).

Let 0, be a constant and we define the norm

is a non-empty convex, closed, and bounded set in Y, .

Then, we show that P maps A into A. Let A.

By using (5), we can get

in which the sequence zn is the corresponding minimal or maximal point of i (i 1, 2). The proof completes.

CONCLUSION

We calculate the existence of nonnegative solution for Leslie-Gower predator-prey model by constructing appropriate upper-lower solution and employing fixed-point theorems. The conditions for the existence of front type the Leslie- Gower formulation are established. Our analysis demonstrate that climate change speed models as a shifting environment. Specifically, it is shown that: Front-type forced waves emerge when populations respond to the environmental shifts by forming monotone traveling wave profiles, capturing the invasion-extinction transition.

.

ACKNOWLEDGMENT

The author would like to express unlimited gratitude to his foremost, respectable and noble teacher, Associate Professor Dr. Aung Zaw Myint, Department of Mathematics, University of Mandalay for his encouragements on doing research. The author acknowledges the unknown reviewers for their suggestions and patient review on this papere.

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