Analysis of Mathematical Ecology and Population Model by Introducing a Prey-Predator Model

Abstract

The purpose of this paper is to develop a mathematical concept for preventing the spread of infectious diseases and other species by studying organisms and their activities in the environment using mathematical ecology and population models. Additionally, we aim to study the fear effect in prey-predator systems by analyzing the fundamental concepts of the model.

Keywords

Mathematical ecology, prey-predator model, population model, least square method, group averages method.

Introduction

Ecology focuses on how predators affect hunting based on population density and hunting-predator interaction.

There are two main approaches to this interaction. In the direct approach, predators tend to kill their prey, viewing the prey as a resource. In the indirect approach, prey change their behavior to avoid being hunted, driven by a sense of danger.

In the prey-predator system, the indirect impact on prey populations, known as the fear effect, is now recognized to be stronger than the direct impact. Prey animals can move to safer areas to avoid predation, even sacrificing their highest intake rate areas.

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This short-term survival strategy can reduce long-term fitness and reproduction stability. Fear of predation has been shown to reduce reproduction in prey species.

Mathematical modeling is a powerful tool for understanding ecological populations and preventing epidemics. Infectious diseases often spread through interactions between species, emphasizing the need for mathematical models in ecology and epidemiology. Researchers have studied epidemic models, such as those developed by Kermac and McKendrick, to understand and combat diseases.

Organisms' interactions with their environment have a significant impact on ecology and biology, influencing species development, population dynamics, and more.

Mathematical models provide a robust framework for analyzing these biological processes, with various models categorized as deterministic models, including simple differential equations.

Mathematical Model

We will start with a simple homogeneous population model:

Let N_t represent the population of a species at any time t, and the rate of change relative to time in the population is represented by:

dN_t/dt = (b - d)N_t

Where:

b: Per capita birth rate

d: Per capita death rate

The difference (b - d) is known as the intrinsic rate of growth, equal to r:

r = b - d

Thus, the equation can be written as:

dN_t/dt = rN_t

With the initial condition:

N₀ = N_t0

The solution to this equation is given by:

N_t = N₀e^rt

This solution shows exponential growth for positive intrinsic growth rates. To illustrate this, consider the population growth rate of India from 1950 to 2020:

Table 1(a): Population Growth Rate of India from 1950 to 2020

Clearly, the population graph demonstrates the population growth rate of India over time. The data shows that the population growth rate has been continuously decreasing. Using the equation N_t = N₀e^rt, we can calculate the population value at any given time t and the intrinsic growth rate r. This can be done using the Least Square method or Group Averages method for the best fit of the equation.

In the next section, we will discuss a mathematical model that analyzes the effect of a virus on a single species. We assume a population density, virus concentration in the population and environment, and certain rates of reduction and growth.

Mathematical Analysis of Virus Effect

In this section, we analyze the effect of a virus on a single species using a mathematical model. We assume the following parameters:

N_t: Population density

C_t: Concentration of the virus in the population

E_t: Concentration of the virus in the environment

The model satisfies the following conditions:

• There is a virus in the environment that humans receive in their bodies, affecting population mobility.
• Growth rate of the population is affected by the virus.
• Birth rate (f) and death rate (α) are considered as positive constants.

The model can be described as:

dN_t/dt = (bf(N_t, E_t) - αN_t)N_t

With initial conditions:

N₀ = N_t0, C₀ = C_t0, E₀ = E_t0

Where:

• f(N_t, E_t): Reduction rate of virus in the population due to egestion
• m(N_t): Reduction rate of virus in the population due to metabolization process
• g(E_t): Rate of reduction of virus in the environment
• h(t): Extrinsic virus input rate, a continuous and bounded nonnegative function of t
• c(N_t, E_t): Reduction rate of virus in the environment due to its intake by the population

Conclusion

This paper combines mathematical modeling with ecological concepts to analyze population dynamics, including exponential growth and the impact of fear in prey-predator systems. Understanding these dynamics is crucial for managing epidemics and preserving ecosystems. Researchers in ecology and mathematics continue to collaborate to study and prevent the prevalence of infectious diseases and other ecological challenges.

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