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The following reference was used for this example
E.J. Doedel, Lecture Notes on Numerical Anlysis of Bifurcation Problems, Hamburg, March 1997.

Predator-prey model

A basic model for studying the interaction of species in population biology is the so-called predator-prey model. The predator-prey model is a planar system representing the behavior of a population of prey, say fish, and a population of predators, say sharks. People like to fish too, and to prevent overfishing, we enforce a fishing-quota . The equations now become

(1)

Here u₁ represents the number of fish, normalized at a certain saturated quantity, and u₂ is the number of sharks, again normalized. Let us consider the first equation of (1). The term

(2)	3u1(1 - u1)

gives the growth of the fish population. If there is little fish, this term is almost equal to 3u₁, which gives linear growth. However, as the number of fish increases, the fish population may become overcrowded and growth will stop, due to limiting resources and the like. If u₁ becomes larger than one, the growth is negative. Other decreasing effects are sharks hunting for fish, u₁u₂, and people who go fishing, (1 - e^-5u₁). The exponential term tells you that it is hard to catch fish when the population is very small.

The growth term in the second equation is 3u₁u₂. This is similar to (2) where now the limiting resources are given by the fish population u₁ instead of (1 - u₂). The term -u₂ represents death of sharks as they grow old.

The four diagrams below show the change in behavior as increases. The pictures were made with DsTool using the model file predprey.c. Note that the coordinate axes {u₁ = 0} and {u₂ = 0} are invariant; the fish populations can only grow if there is any fish to begin with, and the same holds for the sharks.

= 0 2.6 u2 0 0 u1 1

= 0.5 2.6 u2 0 0 u1 1

= 0.68 2.6 u2 0 0 u1 1

= 0.7081 2.6 u2 0 0 u1 1

Figure 1: Phase plane diagrams of equation (1).

The pictures show that for low fishing quota, the fish and shark population reaches an equilibrium state. In fact, if = 0, the shark population is higher than its normalized value. For slightly bigger fishing quota the stable equilibrium is well within the bounds u₁ = 1 and u₂ = 1. For = 0.68 the populations fluctuate in a periodic manner. As is allowed to grow bigger than 0.7081, both fish and sharks go extinct. The bifurcation diagrams below show how these different types of qualitative behavior change from one to the other as increases.

u1			u2
				u1

Figure 2: The bifurcation diagram on the left shows only the dependence of u₁ on . The picture on the right shows the periodic orbits for different values of .

There are four bifurcation points in Figure 2, two pink stars that are transcritical bifurcations, one pink square which is a Hopf bifurcation, and a pink circle that stands for a saddle-node bifurcation. The first two phase diagrams of Figure 1 correspond to behavior in the region before the first pink star. The phase diagram for = 0.68 is qualitatively the behavior in the region where periodic orbits exists. The periodic orbits are created after the Hopf bifurcation for = 0.67, approximately, and their period increases as increases. The period becomes infinite approximately at = 0.7081. The limiting orbit is a heteroclinic cycle, as is shown in the last picture of Figure 1.

Figure 3: The bifurcation diagram plus an enlargement near the Hopf bifurcation; a three-dimensional view in (, u₁, u₂)-space.

Even though the pictures are computed with AUTO, all equilibrium solutions can be found analytically. Let us first consider = 0. Putting the right-hand-side of equation (1) to 0, we find:

		3u1(1 - u1) - u1u2 = 0			u1 = 0	or	3(1 - u1) = u2
		-u2 + 3u1u2			u2 = 0	or	3u1 = 1.

Hence, there are three equilibria: (u₁, u₂) = (0,0), (1,0), and (1/3, 2). For nonzero the point (0,0) stays an equilibrium. Any other equilibrium should satisfy u₂ = 0 or u₁ = 1/3. Therefore, we find three branches of equilibria that intersect at two branch points, the pink stars in Figure 3:

I:	(u1, u2) = (0,0),
II:	u2 = 0	and
III:	u1 = 1/3	and	u2 = 2 - 3(1 - e-5/3).

To complete our investigation, we now consider the stability of the solution branches, by studying the eigenvalues of the Jacobian matrix at an arbitrary equilibrium:

	J(u1, u2) =		3 - 6u1 - u2 - 5e-5u1		-u1
			3u2		-1 + 3u1

J(0,0) is a diagonal matrix with eigenvalues 3 - 5 and -1. Hence, the origin is unstable for in [0, 3/5) and stable beyond = 3/5. For branch II the Jacobian is an upper triangular matrix so that again the eigenvalues are on the diagonal. This branch does not have stable solutions for positive u₁. The saddle-node bifurcation on this branch is the merging of a saddle and a source.

The eigenvalues for branch III are more complicated to compute. For small the equilibrium is stable. Approximately for = 0.67 these eigenvalues cross the imaginary axis, the Hopf bifurcation. At this moment, the stable equilibrium becomes a source. This source later turns into a saddle as the branch intersects branch II.

Summarizing from a practical point of view, the fish and shark population is in no danger as long as the fishing quota is kept small enough. Approximately at = 0.67 the populations start fluctuating (periodic solutions) wich ultimately leads to a collapse to the origin as grows beyond = 0.7081.

Written by Hinke Osinga
Last modified: Tue Mar 30 11:26:24 1999