Jonathan A. Sherratt, Department of Mathematics, Heriot-Watt University

# Cyclic population dynamics

Predator-prey dynamics have been studied extensively using mathematical models. A common feature of many of these models is the prediction that the populations can cycle for some parameter sets: i.e. the populations of predators and prey do not settle to constant values, but rather, oscillate periodically in time. There is considerable data from field studies and laboratory experiments to support the existence of such population cycles.

# Invasion in cyclic populations

If a small group of predators is introduced into an otherwise spatially uniform population of prey, the predators will tend to invade the prey, leaving behind a mixture of predators and prey. In collaboration with Mark Lewis, Barry Eagan and Andrew Fowler, I have studied the behaviour behind such invasions for cyclic populations. The results are somewhat suprising: the invasion leaves behind spatiotemporal oscillations which fall into one of two categories, either a periodic travelling wave or spatiotemporal irregularity; mixed behaviour also occurs.

The animation illustrates an example of such a mixed case, in which a band of periodic waves is visible immediately behind invasion, but develops into irregular oscillations further back.

# Discrete Models

These animations use a reaction-diffusion model, consisting of couple partial differential equations for predators and prey. However we have shown that the same phenomena occur in couple map lattice, cellular automaton and integrodifferential equation models. The commonality between these various models strongly suggests that the behaviour is intrinsic to the underlying biology. The figure shows a simulation of a couple map lattice model for a parameter set giving a periodic wave behind invasion. The colours represent predator density (white=zero, red=low, blue=high).

# Is it Chaos?

An important question is whether the irregular oscillations seen for some parameter sets are genuinely chaotic. For the case of reaction-diffusion models, I addressed this via a numerical bifurcation study in which a fixed domain is consider, with domain length taken as a bifurcation parameter. This reveals periodic doubling and other well-known routes to chaos, as domain length is increased.
Click to see a picture of period-doubling in an oscillatory system

# Which Periodic Wave?

Any oscillatory reaction-diffusion system has a family of periodic travelling wave solutions. Therefore the process described above raises a natural question: which member of this family is selected behind invasion. This is an interesting mathematical question, but is also important in applications, as the answer would enable prediction of the speed and amplitude of the wavetrain, and also its stability. (The chaotic behaviour occurs when an unstable wavetrain is selected).

I have answered this question for systems that are close to a supercritical Hopf bifurcation in the reaction kinetics. Roughly, the calculation divides the solution up into three regions: the periodic wavetrain, the invading wave front, and the region in between these two. A solution is calculated in each region, and then matched up to give a final solution.

The work described on this page is discussed in the following papers:

• J.A. Sherratt: Irregular wakes in reaction-diffusion waves. Physica D 70: 370-382 (1994).

• J.A. Sherratt, M.A. Lewis, A.C. Fowler: Ecological chaos in the wake of invasion. Proc. Natl. Acad. Sci. USA 92: 2524-2528 (1995). Click to see the Full paper (PDF)

• J.A. Sherratt: Unstable wavetrains and chaotic wakes in reaction-diffusion systems of lambda-omega type. Physica D 82: 165-179 (1995). Click to see the Full paper (PDF)

• J.A. Sherratt: Periodic travelling waves in a family of deterministic cellular automata. Physica D 95: 319-335 (1996). Click to see the Full paper (PDF)

• J.A. Sherratt: Oscillatory and chaotic wakes behind moving boundaries in reaction-diffusion systems. Dynam. Stab. Systems 11: 303-325 (1996). Click to see the Full paper (PDF)

• J.A. Sherratt, B.T. Eagan, M.A. Lewis: Oscillations and chaos behind predator-prey invasion: mathematical artifact or ecological reality? Phil. Trans. R. Soc. B 352: 21-38 (1997). Click to see the Full paper (PDF)

• J.A. Sherratt: Invasive wave fronts and their oscillatory wakes are linked by a modulated travelling phase resetting wave. Physica D 117: 145-166 (1998). Click to see the Full paper (PDF)

• J.A. Sherratt: Periodic travelling waves in cyclic predator-prey systems. Ecol. Lett. 4 30-37 (2001). Click to see the Full paper (PDF)

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