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Behaviour of a herbivore-plankton continuous interaction model

Elaboration 2016 17 Pages

Mathematics - Applied Mathematics

Excerpt

Contents

Abstract

1 Introduction

2 Mathematical model
2.1 Solving for the equilibrium points
2.2 Investigating the stability of the equilibrium points
2.3 Computer simulations and the qualitative analysis

3 CONCLUSIONS

REFERENCES

HERBIVORE–PLANKTON INTERACTION MODEL

Kingsley Eshun Gyekye

Department of Mathematics and Computer Science

Lobachevsky State University, Nizhni Novgorod, Russia

Abstract

This paper investigates and analyzes the behaviour of a herbivore-plankton continuous model. Two of the equilibrium points are solved analytically while the third equilibrium point is solved with the help of Nullclines phase portrait. The model’s equilibrium points stability and their ecological implications are analyzed and computer simulations are used to exhibit the characteristics of the model.

Keywords: Equilibrium Point, Stability, Herbivore-Plankton Model, Growth Rate.

1 Introduction

Population growth is one of the biological studies not easy to predict as it involves multiple variables some of which are almost impossible to determine. Engineers and researchers have used mathematical modeling and computer simulations to solve and predict many complex problems which would have been difficult to predict (Dym, 2004). Same can be said about the use of similar techniques for solving ecological problems. Modeling and qualitative analysis of population growth is one of the interesting areas in population ecology as it involves the application of discrete, continuous, linear and nonlinear differential equations. It is important in population ecology as it can help predict either increase or decline in population growth rate at any particular point in time. In the case of farming, modeling and analysis of plants can help farmers predict how well their crops will fare under different environmental conditions and even help them to predict future yields. It can also be used to predict whether a particular plant or animal species is on the verge of extinction Rockwood (2006).

In population ecology, single-species population is the simplest to model as it involves few parameters; a general model can be presented as

Abbildung in dieser Leseprobe nicht enthalten

wherer= (birth rate - death rate + immigration rate - emigration rate).

Population growth of such models increases exponentially especially when their habitat have the abundance of resources to support their numbers. In the case of plants; immigration, emigration and death rate is a small rate since the model does not include herbivores that feed on them.

Two-species population models in population ecology involve two discrete or continuous model of two different species either in competition for food or one killing the other for food. Examples of such models are the predator-prey model and the herbivore-plant model. The most popular of this type of model is the Lotka-Volterra predator-prey model.

In 1926, Lotka-Volterra developed a model that described the existence of a particular fish species predating on another fish species in the Adriatic Sea, explaining why there were not consistency in the level of fish catch in the Adriatic Sea (Murray, 2002).

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He made the following assumptions,

1. From equation (3), in the absence of predation, that is[Abbildung in dieser Leseprobe nicht enthalten], the prey grows exponentially without restriction and the result is equivalent to equation (2).
2. In the case of the existence of predation then the prey growth rate is reduced by the [Abbildung in dieser Leseprobe nicht enthalten] term.
3. From equation (4), in the absence of prey that is [Abbildung in dieser Leseprobe nicht enthalten] , the population of the predator decrease exponentially due to the [Abbildung in dieser Leseprobe nicht enthalten] term.
4. Finally, with the existence of prey in equation (4) the population of the predator increases proportionally to the density of the prey.

2 Mathematical model

In reality, it is only poisonous plants or planktons that do not have herbivores to feed on and therefore models of single plant species without herbivores are best considered for studying purposes because even in a population model where there is coexistence of two species, it is difficult to say that, the existence of one may not affect the other as one may be a food for the other or might kill the other without being aware of the effect (Leah, 2005).

We will consider a model that is more precise in describing a population. Herbivores feed on plants or planktons. Herbivores feeding on planktons reduce the population of the planktons and that can even affect the reproduction cycle of the planktons. On the other hand, the existence of a large amount of planktons will boost the population of the herbivores.

We consider a herbivore-plankton interaction model version of the Lotka-Volterra predator-prey model cited in (Hirsch, Smale, and Devaney, 2013).

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2.1 Solving for the equilibrium points

In this section, we solve for the equilibrium points of the algebraic equations (5) and (6) by equating them to zero. That is

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First, we will solve for the solutions of equation (6) and substitute them into equation (5). By equating equation (6) to zero, we get a simple algebraic quadratic equation

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We will only solve for the equilibrium point at and use a graph to find the other positive equilibrium point in the first quadrant of the Cartesian plane.

We substitute into equation (5) and the result is

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Therefore the points are two of the equilibrium points of the systems and there are others which would be investigated using Nullclines phase portrait.

2.2 Investigating the stability of the equilibrium points

We need to determine the stability of the two equilibrium points. First, we need to find the related eigenvalues of the linearization matrix at the equilibrium.

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Details

Pages
17
Year
2016
ISBN (eBook)
9783668265448
ISBN (Book)
9783668265455
File size
568 KB
Language
English
Catalog Number
v323584
Institution / College
Lobachevsky State University of Nizhni Novgorod – Computational Mechanics
Grade
5.0
Tags
behaviour

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Title: Behaviour of a herbivore-plankton continuous interaction model