Lotka-Volterra analysis

Gause (1934) was the first to systematically study the interactions between ciliates and their prey in a closed laboratory environment. The topic was also studied extensively by Volterra (1931) who, together with Lotka (1925), developed early mathematical models. The original Lotka-Volterra analysis considered and /X2 constants, but normally they would depend on their respective substrates. Thus... 

From a mathematical point of view, the onset of sustained oscillations generally corresponds to the passage through a Hopf bifurcation point [19] For a critical value of a control parameter, the steady state becomes unstable as a focus. Before the bifurcation point, the system displays damped oscillations and eventually reaches the steady state, which is a stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [15, 17]. By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in this chapter. The phase plane analysis of two-variable models indicates that the oscillatory dynamics of neurons also corresponds to the evolution toward a limit cycle [20]. A similar evolution is predicted [21] by models for predator-prey interactions in ecology. 

As was noted in Section 2.1.1, the concentration oscillations observed in the Lotka-Volterra model based on kinetic equations (2.1.28), (2.1.29) (or (2.2.59), (2.2.60)) are formally undamped. Perturbation of the model parameters, in particular constant k, leads to transitions between different orbits. However, the stability of solutions requires special analysis. Assume that in a given model relation between averages and fluctuations is very simple, e.g., (5NASNB) = f((NA), (A b)), where / is an arbitrary function. Therefore k in (2.2.67) is also a function of the mean values NA(t) and NB(t). Models of this kind are well developed in population dynamics in biophysics [70], Since non-linearity of kinetic equations is no longer quadratic, limitations of the Hanusse theorem [23] are lifted. Depending on the actual expression for / both stable and unstable stationary points could be obtained. Unstable stationary points are associated with such solutions as the limiting cycle in particular, solutions which are interpreted in biophysics as catastrophes (population death). Unlike phenomenological models treated in biophysics [70], in the Lotka-Volterra stochastic model the relation between fluctuations and mean values could be indeed calculated rather than postulated. 

Analysis of equations for second momenta like (SNA5NB), (5Na)2) and (5NB)2) shows that all their solutions are time-dependent. In the Lotka-Volterra model second momenta are oscillating with frequencies larger than that of macroscopic motion without fluctuations (2.2.59), (2.2.60). Oscillations of k produce respectively noise in (2.2.68), (2.2.69). Fluctuations in the Lotka-Volterra model are anomalous second momenta are not expressed through mean values. Since this situation reminds the turbulence in hydrodynamics, the fluctuation regime in this model is called also generalized turbulence [68]. The above noted increase in fluctuations makes doubtful the standard procedure of the cut off of a set of equations for random values momenta. 

Statement 1. Provided K t) = K = const, i.e., neglecting change in time of the correlation functions, equations (8.2.12) and (8.2.13) of the concentration dynamics describe undamped concentration oscillations with the frequencies uj < uq = /a, dependent on the initial conditions. The dependence u = uj K) is weak. This statement is based on the analysis of the Lotka-Volterra model by both topological and analytical methods (see Section 2.1.1). 

In the next few sections we ll consider some simple examples of phase plane analysis. We begin with the classic Lotka-Volterra model of competition between two species, here imagined to be rabbits and sheep. Suppose that both species are competing for the same food supply (grass) and the amount available is limited. Furthermore, ignore all other complications, like predators, seasonal effects, and other sources of food. Then there are two main effects we should consider ... 

This analysis suggests that the effect of transport into the system stabilizes the behavior of the equations and in particular allows the solutions to achieve a constant solution. This is in marked contrast to the behavior of the classical Lotka—Volterra equations. 

Fig. 15.3. Pheise portrait of the modified Lotka-Volterra models in the (R, Ai)-phase plane (solid lines). Graphical analysis revolves around plotting the isoclines in the prey-predator phase plane that denote zero-growth of the model predator and prey populations [13] (dashed lines), a) Nonlinear density dependence g R) leads to a decreasing prey isocline and is stabilizing, b) Type-II functional response f R) gives rise to an increasing prey isocline and is destabilizing.

The analysis of the behaviour of the Lotka-Volterra model leads us to investigate fluctuations near an instability point, and we return to this point in Section 5.6. 

What follows from the stochastic analysis of the system In the case of Y = 0 the only stationary state, analogously to the Lotka-Volterra case, is the absorbing state 0, i.e. 

A numerical tool, sensitivity analysis, which can be used to study the effects of parameter perturbations on systems of dynamical equations is briefly described. A straightforward application of the methods of sensitivity analysis to ordinary differential equation models for oscillating reactions is found to yield results which are difficult to physically interpret. In this work it is shown that the standard sensitivity analysis of equations with periodic solutions yields an expansion that contains secular terms. A Lindstedt-Poincare approach is taken instead, and it is found that physically meaningful sensitivity information can be extracted from the straightforward sensitivity analysis results, in some cases. In the other cases, it is found that structural stability/instability can be assessed with this modification of sensitivity analysis. Illustration is given for the Lotka-Volterra oscillator. 

Sensitivity analysis thus provides a method by which the structural stability of multi-parameter models can be assessed and described in more detail. We can say for the reference solution studied here that the Lotka-Volterra oscillator is structurally unstable to variations in I<4 and I<5 but not to variations in k, k2 and I<3. These properties of the Lotka-Volterra oscillator are, of course, well-known. The success of sensitivity analysis in unambiguously (and quantitatively) verifying these facts suggests that it will be a useful tool for the study of models which are not so well-understood. 

Use the data of Prob. 5.7 to fit the Lotka-Volterra predator-prey equations (shown below) in order to obtain accurate estimates of the parameters of the model. Modify the Lotka-Volterra equations as recommended in Prob. 5.8, and determine the parameters of your new models. Compare the results of the statistical analysis for each model, and choose the set of equations that gives the best repre.senration of the data. 

Until the 1950s, the rare periodic phenomena known in chemistry, such as the reaction of Bray [1], represented laboratory curiosities. Some oscillatory reactions were also known in electrochemistry. The link was made between the cardiac rhythm and electrical oscillators [2]. New examples of oscillatory chemical reactions were later discovered [3, 4]. From a theoretical point of view, the first kinetic model for oscillatory reactions was analyzed by Lotka [5], while similar equations were proposed soon after by Volterra [6] to account for oscillations in predator-prey systems in ecology. The next important advance on biological oscillations came from the experimental and theoretical studies of Hodgkin and Huxley [7], which clarified the physicochemical bases of the action potential in electrically excitable cells. The theory that they developed was later applied [8] to account for sustained oscillations of the membrane potential in these cells. Remarkably, the classic study by Hodgkin and Huxley appeared in the same year as Turing s pioneering analysis of spatial patterns in chemical systems [9]. 

Hofbauer, J. (1981). On the recurrence of limit cycles in the Volterra-Lotka equation, Nonlin. Analysis, Theory, Methods and Applications, 5, 1003-7. 

The model in (2.56) has been investigated by GRIFFITH (1968), HUNDING (1974) and TYSON (1975). In contrast to the excitation models in Sections 2.4 and 2.5, it shows a negative feedback a sudden increase of S leads to stepwise delayed increases of the chain products S 2. .. until finally S reduces the enzyme E involved in the first step such that now the chain products Sp S2,. .. S will decrease, etc. From this qualitative consideration we expect that the system will be able to perform oscillatory changes of its state. We shall convince ourselves by a mathematical analysis that this will really be possible however, we will also learn that these oscillatory motions are basically different from those of the undamped Volterra-Lotka model. 

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