Sustainability In Single Species Population Models

Single species population models, Mathematical Models in Population Biology and Epidemiology

Sustainability In Single Species Population Models

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 single species population models with the help of Nullclines phase portrait. 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, Same can be said about the use of similar techniques for solving ecological problems.

Chapter 1.

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.

Malthusian Population Model

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 single species population models. It can also be used to predict whether a particular plant or animal species is on the verge of extinction Rockwood 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.

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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.

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The most popular of this type of model is the Lotka-Volterra predator-prey model. InLotka-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, Abbildung in dieser Leseprobe nicht enthalten He made the following assumptions, 1.

Inhaltsverzeichnis

From equation 3in 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.

In the case of the existence of predation then the prey growth rate is reduced by the [Abbildung in dieser Leseprobe nicht enthalten] term.

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From equation 4in 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. Finally, with the existence of prey in equation 4 the population of the predator increases proportionally to the density of the prey.

We will consider a model that is more precise in describing a population.

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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.

Single species population models

We consider a herbivore-plankton interaction model version of the Lotka-Volterra predator-prey model cited in Hirsch, Smale, and Devaney, Abbildung in dieser Leseprobe nicht enthalten 2. That is Abbildung in dieser Leseprobe nicht enthalten First, we will solve for the solutions of equation 6 and substitute them into equation 5.

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By equating equation 6 to zero, we get a simple algebraic quadratic equation Abbildung in dieser Leseprobe nicht enthalten We will only single species population models for the equilibrium point at and immer wieder flirten a graph to find single species population models other positive equilibrium point single species population models the first quadrant of the Cartesian plane.

We substitute into equation 5 and the result is Abbildung in dieser Leseprobe nicht enthalten Therefore the points are two of the equilibrium points of the systems and there are others which would be investigated using Nullclines phase portrait. First, we need to find the related eigenvalues of the linearization matrix at the equilibrium. Abbildung in dieser Leseprobe nicht enthalten [