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A theoretical model developed by Stone describing a three-level trophic system in the Ocean is analysed. The system consists of two distinct predator-prey networks, linked by competition for nutrients at the lowest level. There is also an interaction at the level of the two preys, in the sense that the presence of one is advantageous to the other when nutrients are low. It is shown that spontaneous oscillations in population numbers are possible, and that they result from a Hopf bifurcation. The limit cycles are analysed using Floquet theory and are found to change from stable to unstable as a solution branch is traversed.

In a recent paper, Stone [

Much research has been performed on the stability of interacting populations in biological models (e.g., see May [

There has been much work done on the effect of interaction functions in dynamical systems. It has been shown by Gross et al. [

Other dynamical structures such as quasiperiodicity have also been found in higher dimension food webs similar to the one being examined here. Ruan [

In the present paper, we use methods from Dynamical Systems theory to analyse the three-level trophic food web described by Stone [

The model is presented in Section

The trophic web considered now is that illustrated in Figure

Stone’s Compartmental model showing interaction between components. The direction of the arrows indicates a direct positive influence by one component on another. Here

The quantity

In their paper, Hadley and Forbes [

The system (

There are five nondimensional parameter groupings in the system (

We now look for the steady-state solutions

When the time-dependent populations are close to any of the four steady-states in (

We are only interested here in the last steady-state

The Hopf curve: The curve shows the location in

Both the supercritical and subcritical limit cycles emerge to the right in Figure

Shertzer et al. [

We used MATLAB to solve the nonlinear system at the parameter values discussed in the previous section. Once we were satisfied that apparently oscillatory solutions were present then, using a shooting algorithm based on Newton’s method, we were able to find if the resultant solutions were in fact periodic limit cycles. The method used will now be described briefly here.

We rescaled (

Newton’s method was used to adjust the estimates of

After we have calculated a limit cycle, we then perform a linear perturbation to the system (

Figure

The amplitude of the limit cycles of

Figure

The dashed lines in Figure

As can be seen from Figure

The period of the limit cycles in Figure

We now look at two distinct solutions to the system occurring at the same value of

Figure

The unstable (dashed line) and stable (solid line) limit cycles for the solution

The eigenvalues of the stable limit cycle displayed on the unit circle.

The eigenvalues of the unstable limit cycle displayed on the unit circle.

Figure

The instability of this solution means that as a result of a small perturbation

The stable limit cycle (red) and the perturbation solutions (blue) for perturbations above and below the limit cycle.

The unstable limit cycle (red) and the perturbation solutions (blue) for perturbations above and below the limit cycle.

Figures

This paper presented an investigation of the solutions to the system proposed by Stone [

Hadley and Forbes [

The periods of the self-sustained oscillations found in this investigation are typically of the order of half a month in dimensional values. This is significantly different to the case in which nutrient supply is constant; for that (degenerate) case, Hadley and Forbes [

It should also be noted that Figure

In Section

We have chosen a Michaelis-Menten term for the nutrient uptake. There are of course other possibilities, although any law that limits the rate for large nutrient concentration might reasonably be expected to behave at least qualitatively similarly to the results presented here. We have not considered migration or alternative models for the interaction between the bacteria and phytoplankton. Nevertheless, this study has shed some light on the dynamics of this system. In addition, we have assumed that the mass of nutrient per organism is roughly constant. The dynamics of this system may possibly become more elaborate if this assumption were to be relaxed. However, these are considerations for further study.

Comments by an anonymous Referee are gratefully acknowledged.