Population Growth

Homework

程熹 2013301020038 物基一班

Abstract

This article discusses the population growth with Euler methods.By changing the initial condition and the value of birth rate and death rate, we can picture different population-time diagram and discuss their difference and reaso

Key Word

Population growth model,euler method.

Introduction

A population model is a type of mathematical model that is applied to the study of population dynamics.Models allow a better understanding of how complex interactions and processes work. Modeling of dynamic interactions in nature can provide a manageable way of understanding how numbers change over time or in relation to each other. Ecological population modeling is concerned with the changes in population size and age distribution within a population as a consequence of interactions of organisms with the physical environment, with individuals of their own species, and with organisms of other species (biophysical env.). Population models are used to determine maximum harvest for agriculturists, to understand the dynamics of biological invasions, and have numerous environmental conservation implications. Population models are also used to understand the spread of parasites, viruses, and disease. The realization of our dependence on environmental health has created a need to understand the dynamic interactions of the earth’s flora and fauna. Methods in population modeling have greatly improved our understanding of ecology and the natural world.
The rate at which a population increases in size if there are no density-dependent forces regulating the population is known as the intrinsic rate of increase.
Where {\displaystyle dN/dt} dN/dt is the rate of increase of the population, N is the population size, and r is the intrinsic rate of increase. This is therefore the theoretical maximum rate of increase of a population per individual. The concept is commonly used in insect population biology to determine how environmental factors affect the rate at which pest populations increase. See also exponential population growth and logistic population growth

Main Features

We use the Euler method to translate the origin equation

$$\frac{dN}{dt}=aN-b N^{2}$$ to $$N(i+1)=N(i)+ aN(i)dt-bN ^{2} (i)dt$$
we can use this equation to design a program to calculate the population N changing according to time T,by set different initial condition and different a and b,we can get completely different population growth curve.

Result

1.For N=100.a=1,b=0.002

In this situation, the population grow to the maximum capicity of this environment and then become stable.

2,For N=100,a=1,b=0.001

Compare to the first situation, as the death rate is lower, the maximum population capicity is greater than the first circumstance,and the rate the population grow is faster than the first situation.

3,For N=1000,a=1,b=0.001

This is the situation that the initial population is greater than the maximum population capicity of the environment, the at the beginning the population will continue, and after it reach the maximum population capacity of the environment, it become to stay stable.

4.For N=100,a=1,b=0

This is the situation when there is no death rate, so the population of the whole system will continue to grow in an incredibley fastly way.

Acknowledgement

This program is finished by myself,but for the value of a and b, I refer to Wangshixing's report.

Reference

Computational Physics(second edition) Nicholas J.Giordano and Hisao Nakanishi
Wikipedia
https://en.wikipedia.org/wiki/Population_dynamics