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The previous section developed techniques for the qualitative analysis of nonlinear differential equations, where the differential equation contained a single unknown variable. The techniques allowed the reader to understand the behavior of mathematical models without actually solving the differential equations. This section looks at a predator-prey model where two species are involved. Thus, the differential equations describing the population dynamics must have two unknown variables, creating a system of differential equations. This system of differential equations cannot be solved exactly, so either numerical or qualitative methods need to be employed. Our studies over the next few sections will develop the techniques to handle these more complicated models.

In this section, we develop the mathematical models for two species that are intertwined in a predator-prey or host-parasite relationship. As we have done before, the differential equations will be analyzed to find equilibria. In subsequent sections, we will discuss the numerical solutions of these systems of differential equations and the stability of the equilibria. These techniques allow the reader to examine a wide range of models that have been developed in biology.

Our study begins with a classical example shown in many ecological textbooks, the population dynamics of a simple predator-prey system. Most mammalian predators rely on a variety of prey, which complicates mathematical modeling; however, a few predators have become highly specialized and seek almost exclusively a single prey species. An example of this simplified predator-prey interaction is seen in Canadian northern forests, where the populations of the lynx and the snowshoe hare are intertwined in a life and death struggle.

One reason that this particular system has been so extensively studied is that the Hudson Bay company kept careful records of all furs from the early 1800s into the 1900s. It is often assumed that the furs are representative of the populations in the wild because of the trapping intensity and trapping techniques, so these data can be used to represent the relative populations of the lynx and hares in the wild. The records for the furs collected by the Hudson Bay company showed distinct oscillations (approximately 12 year periods), suggesting that these species caused almost periodic fluctuations of each other's populations. Below is a graph of some of the fur data from the records of the Hudson Bay company.

We would like to create a mathematical model that explains these population fluctuations. An earlier Lab assignment showed that the data can be readily fit with Fourier series using several trigonometric functions, but fitting data does not give insight into the population dynamics. Ecologists have predicted that in a simple predator-prey system that a rise in prey population is followed (with a lag) by a rise in the predator population. When the predator population is sufficiently high, then the prey population begins dropping. After the prey population falls, then the predator population falls, which allows the prey population to recover and completes one cycle of this interaction. Thus, we see that qualitatively oscillations occur. Can a mathematical model predict this? What causes cycles to slow or speed up? What affects the amplitude of the oscillation or do you expect to see the oscillations damp to a stable equilibrium? The models tend to ignore factors like climate and other complicating factors. How significant are these?

The Predator-Prey Model (Lotka-Volterra)

The predator-prey model was developed in the 1920s by Vito Volterra to explain some fish studies of his son-in-law Humberto D'Ancona. These studies are presented in more detail below with an extended version of the classical predator prey model. Concurrently, Alfred J. Lotka, an American biologist and actuary, produced many of the same models independent of Volterra. Several of the two species predator-prey and competition models are called Lotka-Volterra Models in honor of this biologist and mathematician who developed the basic model. Their models were natural extensions of the Verhulst or logistic model that we studied in the previous section. This work is presented in most basic ecology textbooks and has many references on the web. One good study is shown in the work of M. Beals, L. Gross, and S. Harrell in their web modules from the University of Tennessee. Below we present the basic assumptions needed to create the classical Lotka-Volterra predator-prey model for the dynamics of the populations of a predator and its prey species.

Let H(t) be the population of snowshoe hares and L(t) be the population of lynx. As we have done before, we want to develop a mathematical model based on the growth rates for the populations. That is, the rate of change in a population is equal to the net increase (births) into the population minus the net decrease (deaths) of the population. We begin by applying this to the population of snowshoe hares. The rate of change of the hare population is dH/dt. The primary growth in the hare population is Malthusian, so this says that the population grows in proportion to its own population or a₁H(t). We will make the simplifying assumption that the primary loss of hares is due to predation by lynx. Predation is often modeled by assuming random contact between the species in proportion to their populations with a fixed percentage of those contacts resulting in death of the prey species. Mathematically, this is given by a negative term, -a₂H(t)L(t). Clearly, if there are other major predators or if the lynx population is low such that starvation due to overcrowding dominates the death rate, then alternative death terms would be more appropriate. Combining these terms, we obtain the growth model for the hare population:

Next we examine the population dynamics of the lynx. The rate of change of the lynx population is dL/dt. The primary growth for the lynx population depends on sufficient food for raising lynx kittens, which implies an adequate source of nutrients from predation on hares. Thus, the growth of the lynx population is similar to the death rate for the hare population with a different constant of proportionality. Mathematically, it follows that the growth of the lynx population can be written, b₂H(t)L(t). The loss of lynx is presumed to be a type of reverse Malthusian growth. That is, in the absence of hares, the lynx population declines in proportion to their own population, which mathematically is given by the negative modeling term, -b₁L(t). Clearly, if the lynx diet used other animals or crowding factors from other predators and lynx were taken into account, then these terms would have to be significantly modified. The growth model for the lynx population combines these terms above to give the following:

Clearly, the growth equations above are caricatures of the true complicated dynamics of these two animals. The model ignores the role of climate variation and the interactions of other species, including human disturbance. Other significant factors that are ignored in this modeling effort are the ages of the animals and the spatial distribution. However, we would like to see if this basic model can give a qualitatively similar picture to the dynamics that are observed in the Hudson Bay company furs that are graphed above. The Lotka-Volterra model is given by the system composed of the two differential equations written above. Note that these differential equations are intertwined into a system of differential equations with each growth model depending on the unknown variable (population) of the other.

As in previous analyses of dynamical systems, the first step to a qualitative analysis is finding the equilibria of the model. Equilibrium solutions are ones that do not change in time, so the derivatives are equal to zero. Thus, for the predator prey model above, we set

This results in a system of nonlinear algebraic equations to solve. If we let H_e and L_e be the equilibrium solutions for the snowshoe hare and lynx populations respectively, then the system of algebraic equations that we need to solve is given by

In general, the solution of a system of nonlinear equations can be very difficult, but this case is relatively simple to solve using standard tools from algebra. These two equations can be factored, which aids in finding the solution. The first equation can be written

which implies that either H_e = 0 or a₁-a₂L_e = 0. Thus, we have possible equilibria at

which implies that either L_e = 0 or -b₁+ b₂H_e = 0. Thus, we have possible equilibria at

The simultaneous solution of these two equations shows that when H_e = 0, then L_e = 0, which gives rise to the trivial solution (extinction of both species)

Unfortunately, the equilibria do not help explain the oscillatory behavior of the data reflected in the lynx and snowshoe hare data. We need information about the stability of these equilibria and further numerical and qualitative analysis techniques before we can demonstrate that this is an appropriate model. Below we turn to the original example of Vito Volterra to show how even the equilibrium analysis of this simple model can explain some important ideas in population dynamics of a predator prey system.

Model of Fishing

Humberto D'Ancona was an Italian biologist who, in 1924, completed a statistical study of fish populations in the Adriadic Sea. He observed that, during the time of reduced fishing in World War I, his populations showed an increasing percentage of predator fish, especially sharks and skates (or a decrease of prey fish). This increase in population of the predator fish declined after the war. Vito Volterra, his father-in-law, was an Italian mathematician who had recently retired, and D'Ancona asked Volterra if there was a mathematical model which could explain this observed relative change in the populations of fish species. Within a couple of months, Volterra produced a series of mathematical models for the interaction of two or more species. Below is a table showing his data with a graph.Why should World War I affect the relative frequency of fish in Italian ports?

Percentages of predators in the Fiume fish catch

1914	1915	1916	1917	1918	1919	1920	1921	1922	1923
12	21	22	21	36	27	16	16	15	11

D'Ancona asked his father-in-law Vito Volterra, a famous Italian mathematician, if there was a mathematical model that could explain the increase in predator fish during the World War I period. Within a couple of months, Volterra produced a series of models for the interaction of two or more species.

The data above do not show oscillations, but there is clearly a rise and fall of the percent of selachians (sharks and skates) in the fish catch due to effects of the war. A slight modification of the predator prey model developed above and an equilibrium analysis can be used to explain the observed data. Volterra reasoned that the change in the relative frequency must be related to the changes in the fishing practices during the war. In particular, because of the dangers of fishing during wartime and because of the loss of fishermen who were required to fight in the war, there was a significant decline in the amount of fishing during the war. This needs to be included in the mathematical model for the interaction between the predatory fish and their prey fish.

Let F(t) be the food fish or prey fish and S(t) be the shark and skate population (less desirable catch in those times). The Lotka-Volterra predator-prey model, which includes fishing can be written:

The terms with the coefficients a₁, a₂, b₁, and b₂ are the same as were developed for the predator-prey model for the lynx and hares above. There is another significant predator to both of these populations, human fishermen. The fishermen trolled for these fish primarily with nets that indiscriminately extracted both food fish and the predatory fish (though possibly at different rates). To model the effects of fishing, we add the terms a₃F and b₃S, where a₃ and b₃ reflect the intensity of fishing. These terms are appropriate as an approximation, since the fishing nets will sweep out a percentage of all fish in their paths that is proportional to the populations of fish in the region that is being fished.

Since the data that D'Ancona collected represent annual percentages of fish in the markets, one can assume that the populations of fish during that time are approximately in equilibrium. It can be shown (mathematically) that the average population about a cycle of the Lotka-Volterra model is its equilibrium value. Thus, if the fishing populations are cycling less than annually, the annual catch should reflect the equilibrium population. So this model is assuming a more rapid dynamics of the fish population than the lynx and hares above. If F_e and S_e represent the equilibrium populations of the food fish and sharks/skates populations respectively, then these equilibria are found by solving

As before, these equations are factored to make solving them easy. The first equation is written

which implies that either F_e = 0 or a₁- a₂S_e -a₃ = 0. Thus, the equilibria have either

which implies that either S_e = 0 or -b₁+b₂F_e - b₃ = 0. Thus, we have possible equilibria at

Thus, as before, one of the equilibria is the trivial solution with neither species existing,

which agrees with the equilibrium of the lynx-hare model above. This equilibrium value is fixed and independent of human interaction. It is determined solely by the ecological interactions of the fish and their predators, the sharks and skates. The human fishing activity is reflected in the parameters a₃ and b₃, which are both positive. As the level of fishing increases, these parameters increase. From the analysis above, we saw that the nonzero equilibrium is given by

The equilibrium value of the food fish, F_e, clearly increases as b₃ increases. In one respect, this makes sense because one would expect the food fish to benefit from the harvesting by humans of their natural predators in the Adriadic Sea. However, the model shows that human interference (at least when not too extreme) by harvesting the food fish, which is reflected in the constant a₃, does not appear in the formula for F_e, and so has no effect on the food fish equilibrium. This result is not so intuitive!

Similarly, the equilibrium value of the sharks and skates, S_e, clearly decreases as a₃ increases. This makes sense because the fisherman are in direct competition with the sharks and skates for this food source. However, again the model shows that human interference (at least when not too extreme) by harvesting the sharks and skates, which is reflected in the constant b₃, has no effect on the shark and skate equilibrium. Once again, this result is not so intuitive!

The data of D'Ancona agrees qualitatively with our equilibrium analysis. During World War I, the fishing fleets would be less likely to go out. This implies that the level of fishing decreases, so a₃ and b₃ decrease. Thus, the equilibrium value for the food fish decreases, while the equilibrium for the sharks and skates increases.The result appearing in the market data would be an increase in the percentage of shark and skates available, which is shown in the data. Because of this decrease in the level of fishing, the overall amount of fish products in the market would decrease, which is not shown in the data. There is insufficient data to obtain more than this gross qualitative overview of the effect of human fishing on the population dynamics. Detailed studies would be required to obtain more quantitative results and fit the six parameters in this system of differential equations.

[1] Lotka, A. J. 1925. Elements of Physical Biology. Baltimore: Williams & Wilkins Co.

[2] Volterra, V. 1926. Variazioni e fluttuazioni del numero d'individui in specie animali conviventi. Mem. R. Accad. Naz. dei Lincei. Ser. VI, vol. 2.