Lotka’s system

The system of ODE \dot x=a-bxy, \dot y=bxy-cy was considered in 1910 by Lotka as a model for oscillatory chemical reactions (J. Phys. Chem. 14, 271). It exhibits damped oscillations but no sustained oscillations, i.e. no periodic solutions. It should not be confused with the famous Lotka-Volterra system for predator-prey interactions which was first written down by Lotka in 1920 (PNAS 6, 410) and which does have periodic solutions for all positive initial data. That the Lotka system has no periodic solutions follows from the fact that x^{-1}y^{-1} is a Dulac function. In other words, if we multiply the vector field defined by the right hand sides of the equations by the positive function x^{-1}y^{-1} the result is a vector field with negative divergence. This change of vector field preserves periodic orbits and it follows from the divergence theorem that the rescaled vector field has no periodic orbits. My attention was drawn to this system by the paper of Selkov on his model for glycolysis (Eur. J. Biochem. 4, 79). In his model there is a parameter \gamma which is assumed greater than one. He remarks that if this parameter is set equal to one the system of Lotka is obtained. Selkov obtains his system as a limit of a two-dimensional system with more complicated non-linearities. If the parameter \gamma is set to one in that system equations are obtained which are related to Higgins’ model of glycolysis. Selkov remarks that this last system admits a Dulac function and hence no sustained oscillations and this is his argument for discarding Higgins’ model and replacing it by his own. The equations are \dot x=a-b\frac{xy}{1+y+xy} and \dot y=b\frac{xy}{1+y+xy}-cy. (To obtain the limit already mentioned it is first necessary to do a suitable rescaling of the variables.) In this case the Dulac function is \frac{1+y+xy}{xy}.

The fact that the Lotka-Volterra system admits periodic solutions can be proved by exhibiting a conserved quantity. At this point I recall the well-known fact that while conserved quantities and their generalizations, the Lyapunov functions, are very useful when you have them there is no general procedure for finding them. This naturally brings up the question: if I did not know the conserved quantity for the Lotka-Volterra system how could I find it? One method is as follows. First divide the equation for \dot y by that for \dot x to get a non-autonomous equation for dy/dx, cheerfully ignoring points where \dot x=0. It then turns out that the resulting equation can be solved by the method of separation of variables and that this leads to the desired conserved quantity.

One undesirable feature of the Lotka-Volterra system is that it has a one-parameter family of periodic solutions and must therefore be suspected to be structurally unstable. In addition, if we consider a solution where predators are initially absent the prey population grows exponentially. The latter feature can be eliminated by replacing the linear growth term in the equation for the prey by a logistic one. A similar term corresponding to higher death rates at high population densities can be added in the equation for the predators but the latter modification has no essential effect. This is a Lotka-Volterra model with intraspecific competition. As discussed in the book ‘Evolutionary Games and Population Dynamics’ by Josef Hofbauer and Karl Sigmund, when this model has a positive steady state that state is globally asymptotically stable. The proof uses the fact that the expression which defines the conserved quantity in the usual Lotka-Volterra model defines a Lyapunov function in the case with intraspecific competition. This is an example of the method of obtaining conserved quantities or Lyapunov functions by perturbing those which are already known in special cases.

It follows from Poincaré-Bendixson theory that the steady states in the Lotka model and the Higgins model are globally asymptotically stable. This raises the question whether we could not find Lyapunov functions for those systems. I do not know how. The method used for Lotka-Volterra fails here because the equation for dy/dx is not separable.

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