Predator-prey equations

The previous two chapters showed that competitive exclusion holds under a variety of conditions in the chemostat and its modifications. In this chapter it will be shown that if the competition is moved up one level - if the competition occurs among predators of an organism growing on the nutrient - then coexistence may occur. The fact that the competitors are at a higher trophic level allows for oscillations, and the coexistence that occurs is in the form of a stable limit cycle. Along the way it will be necessary to study a three-level food-chain problem which is of interest in its own right it is the chemostat version of predator-prey equations. The presentation follows that of [BHWl]. 

Integrate the predator-prey equations for the period 1959 1999 using the above constants and initial conditions and compare the simulation with the actual data. Draw the phase plot of IV, versus Atj, and discuss the stability of these equations with the aid of the phase plot. 

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. 

Programs to integrate predator prey equations require"odeiv"... 

In 1914, F. W. Lanchester introduced a set of coupled ordinary differential equations-now commonly called the Lanchester Equationsl (LEs)-as models of attrition in modern warfare. Similar ideas were proposed around that time by [chaseSS] and [osip95]. These equations are formally equivalent to the Lotka-Volterra equations used for modeling the dynamics of interacting predator-prey populations [hof98]. The LEs have since served as the fundamental mathematical models upon which most modern theories of combat attrition are based, and are to this day embedded in many state-of-the-art military models of combat. [Taylor] provides a thorough mathematical discussion. 

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]. 

Beddington, J.R., Free, C.A. and Lawton, J.H. (1975). Dynamic complexity in predator-prey models framed in difference equations, Nature 255, 58-60. 

The Lotka-Volterra model is often used to characterize predator-prey interactions. For example, if R is the population of rabbits (which reproduce autocatalytically), G is the amount of grass available for rabbit food (assumed to be constant), L is the population of lynxes that feeds on the rabbits, and D represents dead lynxes, the following equations represent the dynamic behavior of the populations of rabbits and lynxes ... 

DRS] S. R. Dunbar, K. P. Rybakowski, and K. Schmitt (1986), Persistence in models of predator-prey populations with diffusion, Journal of Differential Equations 65 117-38. 

Pl-lSx (a) There are initially 500 rabbits (x) and 200 foxes (y) on Farmer Oat s property, Use POLYMATH or MATLAB to plot the concentration of foxes and rabbits as a function of time for a period of up to 500 days. The predator-prey relationships are given by the following set of coupled ordinary differential equations ... 

The classical theory of predator-prey interaction as formulated by Volterra involves two equations which express the growth rate of the prey and the predator (57). Within the context of phytoplankton and zooplankton population, the prey is the phytoplankton and the predator the zooplankton. In the notation of the previous sections, for a one-volume system, the Lotka-Volterra equations are ... 

This solution reduces to the previous situation, Equation 40, for large Kmp. This is expected since for KmP large with respect to P, the expression Kmp/(P + Kmp) approaches one. However, an interesting modification from classical predator-prey behavior occurs if the following condition is met... 

The solutions of (3.68)-(3.69) for positive parameters and generic initial conditions are oscillations, of amplitude fixed by the initial conditions, around the fixed point Z — a /a2, P = b2/b. The equation of this family of closed trajectories is ai In Z + b2 In P — biP — a2Z = constant. The oscillations are suggestive of the population oscillations observed in some real predator-prey systems, but they suffer from an important drawback the existence of the continuous family of oscillating trajectories is structurally unstable systems similar to (3.68)-(3.69) but with small additional terms either lack the oscillations, or a single limit cycle is selected out of the continuum. Thus, the model (3.68)-(3.69) can not be considered a robust model of biological interactions, which are never known with enough... 

Equation (5.21) is in fact equivalent to an interesting result which is valid for any reaction occurring on a line with a production which remains proportional to the length of that line. This is the case of the fast binary reaction discussed here, but also of other types such as autocatalytic, competitive, predator-prey, and others to be discussed in the following chapters. When the line becomes folded to form a fractal of dimension Dp (1,2) (at least above a resolution of the order of the product width), we can write (Toroczkai et ah, 1998 Karolyi et ah, 1999 Tel et ah, 2005) ... 

In addition to showing that X changed with respect to H/P, the results indicated also that it depended explictly on time, so that the differential equations describing the predator-prey system are probably non-autonomous. 

Quite clearly, the growth of a predator population is in some way dependent upon the abundance of its prey. The most frequently cited model of predator-prey dynamics is the set of linked, non-linear differential equations known as the Lotka-Volterra equations (1 ). This model assumes that in the absence of predator, the prey grows exponentially, while in the absence of prey the predator dies exponentially, and that the predator growth rate is directly proportional to the product of the prey... 

The fact that bistability is found in such disparate systems as autocataly-tic chemical reaction kinetics and predator-prey dynamics such as that associated with the spruce budworm has led to the concept of normal forms, dynamical models that illustrate the phenomenon in question and are the simplest possible expression of this phenomenon. Physically meaningful equations, such as the reaction rate law for the iodate-arsenite system described above, can, in principle, always be reduced to the associated normal form. Adopting the usual notation of an overhead dot for time differentiation, the normal form for bistability is the following... 

As was mentioned in Subsection 5.6.2, stochastic versions of the Lotka-Volterra model lead to qualitatively different results from the deterministic model. The occurrence of a similar type of results is not too surprising. A simple model for random predator-prey interactions in a varying environment has been studied, staring from generalised Lotka-Volterra equations (De, 1984). The transition probability of extinction is to be determined. The standard procedure is to convert the problem to a Fokker-Planck equation (adopting continuous approximation) and to find an approximation procedure for evaluating the transition probabilities of extinction and of survival. 

Now the question is how to construct the simplest model of a chemical oscillator, in particular, a catalytic oscillator. It is quite easy to include an autocatalytic reaction in the adsorption mechanism, for example A+B—> 2 A. The presence of an autocatalytic reaction is a typical feature of the known Bmsselator and Oregonator models that have been studied since the 1970s. Autocatalytic processes can be compared with biological processes, in which species are able to give birth to similar species. Autocatalytic models resemble the famous Lotka-Volterra equations (Berryman, 1992 Valentinuzzi and Kohen, 2013), also known as the predator-prey or parasite-host equations. 

Let us now consider in detail the classical predator-prey problem, that is, the interaction between two wild-life species, the prey, which is a herbivore, and the predator, a carnivore. These two animals coinhabit a region where the prey have an abundant supply of natural vegetation for food, and the predators depend on the prey for their entire supply of food. This is a simplification of the real ecological system where more than two species coexist, and where predators usually feed on a variety of prey. The Lotka-Volterra equations have also been formulated for such... 

Equations (l)-(3) constitute the complete mathematical formulation of the predator-prey problem based on assumptions (a) to (d). Different assumptions would yield another set of differential equations [see Problem (5.8)]. In addition, the choice of constants and initial conditions influence the solution of the differential equations and generate a diverse set of qualitative behavior patterns for the two populations. Depending on the form of the differential equations and the values of the constants chosen, the solution patterns may vary from stifle, damped oscillations, where the species reach their respective stable symbiotic population densities, to highly unstable situations, in which one of the species is driven to extinction while the other explodes to extreme population density. 

Listing 10.25. Code for integration of predator prey differential equations. 

Although flie predator prey problem has a fairly simple set of nonlinear differential equations, it is typical of the nonlinear eoupling found in real world problems involving many sets of physieal parameters. 

Prey-predator or host-parasite systems can oe analyzed by mass balance equations ... 

Disappearance of predators may also imbalance the equilibrium, and the problem scales up, such as the disappearance of foxes, predators of the deer mouse, which has allowed spreading of the hantavirus in the US, carried by mice (Levins 1993). Similarly, Sabia virus has emerged in Brazil, Guaranito virus in Venezuela (Lisieux 1994), machupo virus in Bolivia, and Junin fever in Argentine (Garrett 1994). In contrast, in a robust ecosystem, elimination of a predator provides space for another predator, such as in the disappearance of the coyote, which has opened the control of field mice to snakes and owls. When both predator and prey are endangered, it may occur that the prey develops resistance. This is taken into account in Volterra s equation (Ehrlich 1986). 

Fig. 2.13. The random trajectory in the stochastic Lotka-Volterra model, equation (2.2.64). Parameters are a/k = /3/k = 20, the initial values NA = NB = 20. When the trajectory coincides with the NB axis, prey animals A are dying out first and predators second.

Fig. 2.16. The random trajectory in the stochastic Lotka model, equation (2.2.76). Parameters are fco//3 = fijk = 10, the initial values Na = Nb = 10. When the trajectory touches the Na axis, the predators B are dying out and the population of the prey animals A infinitely...

From the 1970s onwards, Cesare Marchetti and other system analysts have studied thousands of artifacts, and have discovered that their behaviour is described by the same equations that Lotka and Volterra found for the behaviour of predators and prey. The growth pattern of cars, for example, is a logistic curve. Cars spread in a market exactly as bacteria in a broth or rabbits in a prairie. Cultural novelties diffuse into a society as mutant genes in a population, and markets behave as their ecological niches. But why ... 

---