## Introduction

A major advantage of applying mathematics to ecological systems is the ability to construct mathematical models. Mathematical models are often used for discussing theoretical principles in population dynamics and for drawing conclusions concerning 'qualitative' or 'phenomenological' properties of the dynamics. Such models are mathematical systems that attempt to represent the complex interactions of ecological systems in a way simple enough for their consequences to be understood and explored. This kind of model, however, is restricted by technology as well as technological ingenuity. Another advantage of the mathematical treatment of ecological problems is that it can bring to the surface answers that would have been otherwise overlooked. It is well known that the dynamic behavior of the ecosystem has long been and will continue to be one of the domniant themes in both ecology and mathematical ecology due to its universal existence and importance. For more details, see Model Types: Overview.

Throughout much of ecology's history, most ecologists have believed that ecological systems tend toward an equilibrium - a notion popularized as 'the balance of nature' - and that deviations from that equilibrium are caused by external perturbations. By assuming that systems were close to equilibrium, ecologists could describe them with linear models that were mathematically tractable. However, ecology is rife with processes whose rates depend nonlinearly on the state of the system - such as nutrient uptake, density dependence, and predation.

Today, the concept of stable equilibrium and chaos are commonplace in the scientific community. The term stability is used in ecology to describe the ability of a system to withstand perturbations. For instance, in ecology, asymptotic stability indicates resistance to small perturbations of the state variables. Figure 1 shows a classification scheme of the fixed points of two-dimensional phase spaces.

Stability is a classical subject, whereas chaos is a recent field. There is one class of mechanisms that controls both, namely, bifurcation - the term goes back to Jacobi and was used by Poincare.

In fact, physically, an equilibrium represents a situation without 'life'. The full richness of the nonlinear world det DF

det DF

Figure 1 A classification scheme of the fixed points of two-dimensional phase spaces.

Figure 1 A classification scheme of the fixed points of two-dimensional phase spaces.

is not found at an equilibrium point. On the other hand, though chaos was first knowingly observed in a mathematical model in the early 1960s and it is currently very much in vogue to study chaotic behavior of nonlinear dynamic systems, four decades of investigation suggests that ecological chaos is rare in natural systems. However, the phenomenon of bifurcation has been supported by many experiments, including chemical reaction, driven nonlinear oscillations, Rayleigh-Benard cells, etc.

With the ideas discussed above, this article presents a scientific overview of the basic principles of bifurcation. The subsequent sections give a bifurcation analysis and spatial Turing bifurcation; moreover, these results are discussed in detail. Finally, conclusions and remarks are given in the last section.

## Worm Farming

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