Mathematically speaking, a bifurcation can be defined as a qualitative change in the topology of the phase space of system. It occurs due to a change in the value of the bifurcation parameter resulting in a spontaneous symmetry breaking. In practice, this can be observed as the appearance of a new stationary state or as changes in the stability of the stationary states.

Pseudo-Hopf

Homoclinic

Hopf

Saddle-node

Figure 2 ATakens-Bogdanov (TB, circle) bifurcation is formed by the interaction of Hopf and saddle-node bifurcations. The branch of Hopf bifurcations ends in the TB bifurcation. A branch of homoclinic bifurcations and a branch of pseudo-Hopf situations emerge from the TB bifurcation. From Gross T (2004) Population Dynamics: General Results from Local Analysis. PhD Thesis, Tonning: Der Andere Verlag, Germany.

There are many types of bifurcations, for example, the saddle-node bifurcation (Figures 3 and 4), transcritical bifurcation (Figure 5), pitchfork bifurcation (Figure 6), and Hopf bifurcation (Figure 7). These types differ by the corresponding topological changes in the phase portrait of the system. Consequently, they have different effects on ecological dynamics. Bifurcations may also be classified as subcritical and supercritical (Figure 5) depending on the direction of the bifurcation. A further classification is based on the co-dimension of the bifurcation, which more or less tells the number of parameters that have to be adjusted in order to find the bifurcation point, such as Takens-Bogdanov (TB) bifurcation which is a codimension-2 bifurcation (cf.Figure 2) and Hopf-Turing bifurcation (cf Figure 18).

The term bifurcation refers specifically to the main body of one item splitting into two parts. The term also implies that the item being split is a pathway or avenue for the conveyance of an item or material. For example, a stream, roads, and pipes may all bifurcate; however, a stick which has a main body and splits into smaller items would be less likely to be referred to as having a bifurcation.

If the main item is splitting into three or more parts, more specific terminology can be used, such as trifurca-tion, but such instances are much less likely to occur than bifurcation.

In the following, we illustrate some examples to introduce the basic types of bifurcation.

Figure 3 Saddle-node bifurcation of x — p-x2.

Figure 3 Saddle-node bifurcation of x — p-x2.

to continuous dynamical systems. In discrete dynamical systems, the same bifurcation is often called a fold bifurcation, instead.

Example 1: Consider the one-dimensional system x — p - x2, [1]

where x is the state variable and p is the bifurcation parameter.

If p >0, there are two fixed points, a stable fixed point at - yfp and an unstable one at^fp. At p — 0 (the bifurcation point), there is exactly one fixed point. And the fixed point is no longer hyperbolic. In this case the fixed point is called a saddle-node fixed point. And if p <0 there is no fixed point. The following phase portrait shows the saddle-node bifurcation (Figure 3). The real line represents stable equilibrium points while the broken line represents nonstable points. It is easy to know that the minimum and maximum of p as a function of the curve length denote the saddle-node bifurcation where a stable fixed point (a node) annihilates an unstable one (a saddle in more than one dimension).

The name 'saddle-node' is motivated by the stability behavior of the solutions when they are interpreted as equilibrium of differential equation. The question arises as to whether a bifurcation point always separates stable equilibrium from unstable equilibrium. The answer is no. It is easy to construct a counterexample in two dimensions:

A stability analysis of the equilibrium of system [2] reveals that one equilibrium is a saddle and the other is a node. The stability of the node is determined by the sign c. The result is shown in Figure 4. We learn from this example that both the half-branches meeting at a bifurcation point can be unstable.

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