12.3 a Predator--Prey Model

Giordano-5166 50904_12_ch12_p524-568 January 23, 2013 19:40 539

12.3 A Predator–Prey Model 539

the budworm–forest interaction in an effort to plan for and control the damage. The monograph surveys the ecological situation and examines the computer simulation and differential equation models that are currently in use.

12.3 A Predator--Prey Model

In this section we study a model of population growth of two species in which one species is the primary food source for the other. One example of such a situation occurs in the Southern Ocean, where the baleen whales eat the ﬁsh known as the Antarctic krill, Eu- phausia superboa, as their principal food source. Another example is wolves and rabbits in a closed forest: the wolves eat the rabbits for their principal food source, and the rabbits eat vegetation in the forest. Still other examples include sea otters as predators and abalone as prey and the ladybird beetle, Novius cardinalis, as predator and the cottony cushion insect, Icerya purchasi, as prey.

Problem Identification Let’s take a closer look at the situation of the baleen whales and Antarctic krill. The whales eat the krill, and the krill live on the plankton in the sea. If the whales eat so many krill that the krill cease to be abundant, the food supply of the whales is greatly reduced. Then the whales will starve or leave the area in search of a new supply of krill. As the population of baleen whales dwindles, the krill population makes a comeback because not so many of them are being eaten. As the krill population increases, the food supply for the whales grows and, consequently, so does the baleen whale population. Also, more baleen whales are eating increasingly more krill again. In the pristine environment, does this cycle continue indefinitely or does one of the species eventually die out? The baleen whales in the Southern Ocean have been overexploited to the extent that their current population is approximately one-sixth its estimated pristine level. Thus, there appears to be a surplus of Antarctic krill. (Already, approximately 100,000 tons of krill are being harvested annually.) What effect does exploitation of the whales have on the balance between the whale and krill populations? What are the implications that a krill fishery may hold for the depleted stocks of baleen whales and for other species, such as seabirds, penguins, and fish, that depend on krill for their main source of food? The ability to answer such questions is important to management of multispecies fisheries. Let’s see what answers can be obtained from a graphical modeling approach.

Assumptions Let x.t/ denote the Antarctic krill population at any time t, and let y.t/ denote the population of baleen whales in the Southern Ocean. The level of the krill population depends on a number of factors, including the ability of the ocean to support them, the existence of competitors for the plankton they ingest, and the presence and levels of predators. As a rough first model, let’s start by assuming that the ocean can support an unlimited number of krill so that dx ax for a > 0 dt D (Later we may want to refine the model with a limited growth assumption. This refinement is presented in UMAP 610 described in the 12.1 Projects section.) Second, assume the krill

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540 Chapter 12 Modeling with Systems of Differential Equations

are eaten primarily by the baleen whales (so neglect any other predators). Then the growth rate of the krill is diminished in a way that is proportional to the number of interactions between them and the baleen whales. One interaction assumption leads to the differential equation dx ax bxy .a by/x (12.8) dt D D Notice that the intrinsic growth rate k a by decreases as the level of the baleen whale population increases. The constants a andDb indicate the degrees of self-regulation of the krill population and the predatoriness of the baleen whales, respectively. These coefficients must be determined experimentally or from historical data. So far, Equation (12.8) governing the growth of the krill population looks just like either of the equations in the competitive hunter model presented in the preceding section. Next, consider the baleen whale population y.t/. In the absence of krill the whales have no food, so we will assume that their population declines at a rate proportional to their numbers. This assumption produces the exponential decay equation dy my for m > 0 dt D However, in the presence of krill the baleen whale population increases at a rate proportional to the interactions between the whales and their krill food supply. Thus, the preceding equation is modified to give dy my nxy . m nx/y (12.9) dt D C D C Notice from Equation (12.9) that the intrinsic growth rate r m nx of the whales increases as the level of the krill population increases. The positiD veCcoefficients m and n would be determined experimentally or from historical data. Putting results (12.8) and (12.9) together gives the following autonomous system of differential equations for our predator–prey model: dx .a by/x (12.10) dt D dy . m nx/y dt D C

where x.0/ x0, y.0/ y0, and a, b, m, and n are all positive constants. The system (12.10) governsD the interactionD of the baleen whales and Antarctic krill populations under our unlimited growth assumptions and in the absence of other competitors and predators.

Graphical Analysis of the Model Let’s determine whether the krill and whale populations reach equilibrium levels. The rest points or equilibrium levels occur when dx=dt dy=dt 0. Setting the right side of Equations (12.10) equal to zero and solving for x andD y simultaneouslyD give the rest points .x; y/ .0; 0/ and .x; y/ .m=n; a=b/. Along the vertical line x m=n and the x axis in theDphase plane, the groDwth dy=dt in the baleen whale populationD is zero; along the horizontal line y a=b and the y axis, the growth dx=dt in the krill population is zero. These features areDdepicted in Figure 12.14.

12.3 A Predator–Prey Model 541

J Figure 12.14 y Rest points of the predator–prey model given by the system (12.10)

Whales a m– , – n b –a b dx dy –– = 0 –– = 0 dt dt x (0, 0) m– Krill

n Learning © Cengage Because the values for the constants a, b, m, and n in system (12.10) will be only estimates, we need to investigate the behavior of the solution trajectories near the two rest points .0; 0/ and .m=n; a=b/. Thus, we analyze the signs of dx=dt and dy=dt in the phase plane. In system (12.10), the vertical line x m=n divides the phase plane into two half- planes. In the left half-plane dy=dt is negativDe, and in the right half-plane it is positive. In a similar way, the horizontal line y a=b determines two half-planes. In the upper half-plane dx=dt is negative, and in the lowerD half-plane it is positive. The directions of the associated trajectories are indicated in Figure 12.15. Along the y axis, motion must be vertical and toward the rest point .0; 0/, and along the x axis, motion must be horizontal and away from the rest point .0; 0/. J Figure 12.15 y Trajectory directions in the predator–prey model Whales

–a b

x m– Krill

n Learning © Cengage From Figure 12.15 you can see that the rest point .0; 0/ is unstable. Entrance to the rest point is along the line x 0, where there is no krill population. Thus, the whale population declines to zero in the absenceD of its primary food supply. All other trajectories recede from the rest point. The rest point .m=n, a=b/ is more complicated to analyze. The information the ﬁgure gives is insufﬁcient to distinguish between the three possible motions shown in Figure 12.10 of the preceding section. We cannot tell whether the motion is periodic, asymptotically stable, or unstable. Thus, we must perform a further analysis.

An Analytic Solution of the Model Because the number of baleen whales depends on the number of Antarctic krill available for food, we assume that y is a function of x. Then from the chain rule for derivatives, we have dy dy=dt dx D dx=dt or dy . m nx/y C (12.11) dx D .a by/x

542 Chapter 12 Modeling with Systems of Differential Equations

Equation (12.11) is a separable ﬁrst-order differential equation and may be rewritten as Â a Ã mÁ b dy n dx (12.12) y D x Integration of each side of Equation (12.12) yields

a ln y by nx m ln x k1 D C or

a ln y m ln x by nx k1 C D

where k1 is a constant. Using properties of the natural logarithm and exponential functions, this last equation can be rewritten as yaxm by nx K (12.13) e C D where K is a constant. Equation (12.13) deﬁnes the solution trajectories in the phase plane. We now show that these trajectories are closed and represent periodic motion.

Periodic Predator–Prey Trajectories Equation (12.13) can be rewritten as

Â ya Ã Âenx Ã K (12.14) eby D xm

Let’s determine the behavior of the function f .y/ ya=eby . Using the ﬁrst derivative test (see Problem 1 in the problem set), we can easily shoD w that f .y/ has a relative maximum at y a=b and no other critical points. For simplicity of notation, call this maximum D value My . Moreover, f .0/ 0, and from 1’Hôpital’s rule, f .y/ approaches 0 as y tends to inﬁnity. Similar argumentsD apply to the function g.x/ xm=enx , which achieves its D maximum value Mx at x m=n. The graphs of the functions f and g are depicted in Figure 12.16. D

J Figure 12.16 f(y) g(x) Graphs of the functions f(y ) y a =e by and Mx D m nx My g (x ) x =e m –nx D yae–by x e

y x a – m–

b n Learning © Cengage

a by m nx From Figure 12.16, the largest value for y e x e is My Mx . That is, Equa- tion (12.14) has no solutions if K > My Mx and exactly one solution, x m=n and D y a=b, when K My Mx . Let’s consider what happens when K < My Mx . D D Suppose K sMy;where s < Mx is a positive constant. Then the equation D m nx x e s D

12.3 A Predator–Prey Model 543

J Figure 12.17 g(x) m nx The equation x e s has exactly two solutionsD Mx for s < Mx . xme–nx s

x x x m m– M

n Learning © Cengage

has exactly two solutions: xm < m=n and xM > m=n (Figure 12.17). Now if x < xm, then m nx nx m x e < s so that se x > 1 and

a by nx m nx m f .y/ y e Ke x sMy e x > My D D D

Therefore, there is no solution for y in Equation (12.14) when x < xm. Likewise, there is no solution when x > xM . If x xm or x xM , Equation (12.14) has exactly the one solution y a=b. D D D Finally, if x lies between xm and xM , Equation (12.14) has exactly two solutions. The smaller solution y1.x/ is less than a=b, and the larger solution y2.x/ is greater than a=b. This situation is depicted in Figure 12.18. Moreover, as x approaches either xm or xM , f .y/ approaches My so that both y1.x/ and y2.x/ approach a=b. It follows that the trajectories deﬁned by Equation (12.14) are periodic and have the form depicted in Figure 12.19.

J Figure 12.18 f(y)

When xm < x < xM , there are exactly two solutions for My y in Equation (12.14). yae–by

nx –m sMye x y y1(x) a y2(x)

J Figure 12.19 y Trajectories in the vicinity of the rest point (m=n; a=b ) are periodic. Whales

–a b

x m– Krill

544 Chapter 12 Modeling with Systems of Differential Equations

Model Interpretation What conclusions can be drawn from the trajectories in Fig- ure 12.19? First, because the trajectories are closed curves, they predict that under the assumptions of our Model (12.10), neither the baleen whales nor the Antarctic krill will become extinct. (Remember, the model is based on the pristine situation.) The second obser- vation is that along a single trajectory the two populations fluctuate between their maximum and minimum values. That is, starting with populations in the region where x > m=n and y > a=b, the krill population will decline and the whale population will increase until the krill population reaches the level x m=n, at which point the whale population also begins to decline. Both populations continueD to decline until the whale population reaches the level y a=b and the krill population begins to increase, and so on, counterclockwise around the trajectoryD . Recall from our discussion in Section 12.1 that the trajectories never cross. A sketch of the two population curves is shown in Figure 12.20. In the figure, we can see that the krill population fluctuates between its maximum and minimum values over one complete cycle. Notice that when the krill are plentiful, the whale population has its maximum rate of increase but that the whale population reaches its maximum value after the krill population is on the decline. The predator lags behind the prey in a cyclic fashion.

J Figure 12.20 Krill The whale population lags behind the krill population x as both populations M m– ﬂuctuate cyclically between n their maximum and minimum values. xm

Time Whales Lag

yM –a b ym

Effects of Harvesting For given initial population levels x.0/ x0 and y.0/ y0, the whale and krill populations will ﬂuctuate with time around one Dof the closed trajectoriesD depicted in Figure 12.19. Let T denote the time it takes to complete one full cycle and return to the starting point. The average levels of the krill and baleen whale populations over the time cycle are deﬁned, respectively, by the integrals 1 Z T 1 Z T x x.t/ dt and y y.t/ dt D T 0 D T 0 Now, from Equation (12.8) Â 1 Ã Âdx Ã a by x dt D

12.3 A Predator–Prey Model 545

so that integration of both sides from t 0 to t T leads to D D Z T Â 1 Ã Âdx Ã Z T dt .a by/ dt 0 x dt D 0 or Z T ln x.T / ln x.0/ aT b y.t/ dt D 0 Because of the periodicity of the trajectory, x.T / x.0/, this last equation gives the average value D a y D b In an analogous manner, it can be shown that m x D n (see Problem 2 in the problem set). Therefore, the average levels of the predator and prey populations are in fact their equilibrium levels. Let’s see what this means in terms of harvesting krill. Let’s assume that the effect of fishing for krill is to decrease its population level at a rate rx.t/. The constant r indicates the intensity of fishing and includes such factors as the number of fishing vessels at sea and the number of crew members on those vessels casting nets for krill. Because less food is now available for the baleen whales, assume the whale population also decreases at a rate ry.t/. Incorporating these fishing assumptions into our model, we obtain the refined model dx .a by/x rx Œ.a r/ by x dt D D (12.15) dy . m nx/y ry Œ .m r/ nx y dt D C D C C The autonomous system (12.15) is of the same form as Equations (12.10) (provided that a r > 0/ with a replaced by a r and m replaced by m r. Thus, the new average population levels will be C m r a r x C and y D n D b Consequently, a moderate amount of harvesting krill (so that r < a/ actually increases the average level of krill and decreases the average baleen whale population (under our assumptions for the model). The increase in krill population is beneficial to other species in the Southern Ocean (seals, seabirds, penguins, and fish) that depend on the krill as their main food source. The fact that some fishing increases the number of krill is known as Volterra’s principle. The autonomous system (12.10) was first proposed by Lotka (1925) and Volterra (1931) as a simple model of predator–prey interaction. The Lotka–Volterra model can be modified to reflect the situation in which both the predator and the prey are diminished by some kind of depleting force, such as the application of insecticide treatments that destroy both the insect predator and its insect prey. An example is given in Problem 3 in the problem set.

546 Chapter 12 Modeling with Systems of Differential Equations

Some biologists and ecologists argue that the Lotka–Volterra Model (12.10) is un- realistic because the system is not asymptotically stable, whereas most observable natural predator–prey systems tend to equilibrium levels over time. Nevertheless, regular population cycles, as suggested by the trajectories in Figure 12.19, do occur in nature. Some scientists have proposed models other than the Lotka–Volterra model that do exhibit oscillations that are asymptotically stable (so that the trajectories approach equilibrium solutions). One such model is given by

dx ax bxy rx2 dt D C dy my nxy sy2 dt D C In this last autonomous system, the term rx2 indicates the degree of internal competition of the prey for their limited resource (such as food and space), and the term sy2 indicates the degree of competition among the predators for the ﬁnite amount of available prey. An analysis of this model is more difﬁcult than that presented for the Lotka–Volterra model, but it can be shown that the trajectories of the model are not periodic and tend to equilibrium levels. The constants r and s are positive and would be determined by experimentation or historical data.

12.3 PROBLEMS

1. Apply the first and second derivative tests to the function f .y/ ya=eby to show that y a=b is a unique critical point that yields the relative maximumD f .a=b/. Show also thatDf .y/ approaches zero as y tends to infinity. 2. Derive the result that the average value x of the prey population modeled by the Lotka– Volterra system (12.10) is given by the population level m=n. 3. In 1868 the accidental introduction into the United States of the cottony cushion insect (Icerya purchasi) from Australia threatened to destroy the American citrus industry. To counteract this situation, a natural Australian predator, a ladybird beetle (Novius cardinalis) was imported. The beetles kept the scale insects down to a relatively low level. When DDT was discovered to kill scale insects, farmers applied it in the hope of reducing even further the scale insect population. However, DDT turned out to be fatal to the beetle as well, and the overall effect of using the insecticide was to increase the numbers of the scale insect. Modify the Lotka–Volterra model to reflect a predator–prey system of two insect species where farmers apply (on a continuing basis) an insecticide that destroys both the insect predator and the insect prey at a common rate proportional to the numbers present. What conclusions do you reach concerning the effects of the application of the insecticide? Use graphical analysis to determine the effect of using the insecticide once on an irregular basis. 4. In a 1969 study, E. R. Leigh concluded that the fluctuations in the numbers of Canadian lynx and its primary food source, the hare, trapped by the Hudson’s Bay Company