7.7 LOTKA-VOLTERRA M ODELS

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1 77 LOTKA-VOLTERRA M ODELS sstems, such as the undamped pendulum, enabled us to draw the phase plane for this sstem and view it globall In that case we were able to understand the motions for all initial conditions Unfortunatel, total energ is not a strict Lapunov function for an equilibria of the damped sstem Instead, we compute Ė(v) b 2, so that the inequalit Ė(v) 0 fails to be strict for points arbitraril near the equilibrium Notice that trajectories (other than equilibrium points) do not sta on the -ais, the set on which Ė 0 Therefore we might epect trajectories to behave as if E were a strict Lapunov function The following corollar, often called LaSalle s Corollar to the Lapunov Theorem 723, not onl provides another means of deducing asmptotic stabilit for equilibria of the damped sstem, but also gives information as to the etent of the basin of attraction for each asmptoticall stable equilibrium We postpone the proof of the theorem until Chapter 8 when we stud limit sets of trajectories Corollar 726 (Barbashin-LaSalle) Let E be a Lapunov function for v on the neighborhood W, as in Definition 722 Let Q v W : Ė(v) 0 Assume that W is forward invariant If the onl forward-invariant set contained completel in Q is v, then v is asmptoticall stable Furthermore, W is contained in the basin of v; that is, for each v 0 W, lim t (F(t, v 0 )) v E XERCISE T77 Let ẍ bẋ sin 0, for a constant b 0 Convert the differential equation to a first-order sstem Use Corollar 726 to show that the equilibria (n, 0), for even integers n are asmptoticall stable (c) Sketch the phase plane for the associated first-order sstem 77 LOTKA-VOLTERRA M ODELS A famil of models called the Lotka-Volterra equations are often used to simulate interactions between two or more populations Interactions are of two tpes Competition refers to the possibilit that an increase in one population is bad for the other populations: an eample would be competition for food or habitat On the other hand, sometimes an increase in one population is good for the other Owls are happ when the mouse population increases We will consider two cases of Lotka-Volterra equations, called competing species models and predator-pre models 309

2 D IFFERENTIAL E QUATIONS 30 We begin with two competing species Because of the finiteness of resources, the reproduction rate per individual is adversel affected b high levels of its own species and the other species with which it is in competition Denoting the two populations b and, the reproduction rate per individual is ẋ a( ) b, (746) where the carring capacit of population is chosen to be (sa, b adjusting our units) A similar equation holds for the second population,sothatwehave the competing species sstem of ordinar differential equations ẋ a( ) b ẏ c( ) d (747) where a, b, c, and d are positive constants The first equation sas the population of species grows according to a logistic law in the absence of species (ie, when 0) In addition, the rate of growth of is negativel proportional to, representing competition between members of and members of The larger the population, the smaller the growth rate of The second equation similarl describes the rate of growth for population The method of nullclines is a technique for determining the global behavior of solutions of competing species models This method provides an effective means of finding trapping regions for some differential equations In a competition model, if a species population is above a certain level, the fact of limited resources will cause to decrease The nullcline, a line or curve where ẋ 0, marks the boundar between increase and decrease in The same characteristic is true of the second species, and it has its own curve where ẏ 0 The net two eamples show that the relative orientation of the and nullclines determines which of the species survives E XAMPLE 727 (Species etinction) Set the parameters of (747) to be a, b 2, c, and d 3 To construct a phase plane for (747), we will determine four regions in the first quadrant: sets for which (I) ẋ 0andẏ 0; (II) ẋ 0andẏ 0; (III) ẋ 0andẏ 0; and (IV) ẋ 0andẏ 0 Other assumptions on a, b, c, and d lead to different outcomes, but can be analzed similarl In Figure 78 we show the line along which ẋ 0, dividing the plane into two regions: points where ẋ 0 and points where ẋ 0 Analogousl, Figure 78 shows regions where ẏ 0andẏ 0, respectivel Combining the information from these two figures, we indicate regions (I) (IV) (as described above) in Figure 79 Along the nullclines (lines on which either ẋ 0or

3 77 LOTKA-VOLTERRA M ODELS = 0 > 0 < 0 < 0 > 0 = 0 Figure 78 Method of nullclines for competing species The straight line in shows where ẋ 0, and in it shows where ẏ 0for a, b 2, c, and d 3 in (747) The -ais in is also an -nullcline, and the -ais in is a -nullcline III IV I II Figure 79 Competing species: Nullclines The vectors show the direction that trajectories move The nullclines are the lines along which either ẋ 0orẏ 0 In this figure, the -ais, the -ais, and the two crossed lines are nullclines Triangular regions II and III are trapping regions 3

4 D IFFERENTIAL E QUATIONS Figure 720 Competing species, etinction The phase plane shows attracting equilibria at (, 0) and (0, ), and a third, unstable equilibrium at which the species coeist The basin of (0, ) is shaded, while the basin of (, 0) is the unshaded region One or the other species will die out ẏ 0), arrows indicate the direction of the flow: left/right where ẏ 0 or up/down where ẋ 0 Notice that points where the two different tpes of nullclines cross are equilibria There are four of these points The equilibrium (0, 0) is a repellor; ( 5, 2 5) is a saddle;and (, 0) and (0, ) are attractors The entire phase plane is sketched in Figure 720 Almost all orbits starting in regions (I) and (IV) move into regions (II) and (III) (The onl eceptions are the orbits that come in tangent to the stable eigenspace of the saddle ( 5, 2 5) These one-dimensional curves form the stable manifold of the saddle and are discussed more full in Chapter 0) Once orbits enter regions (II) and (III), the never leave Within these trapping regions, orbits follow the direction indicated b the derivative toward one or the other asmptoticall stable equilibrium The basins of attraction of the two possibilities are shown in Figure 720 The stable manifold of the saddle forms the boundar between the basin shaded gra and the unshaded basin We conclude that for almost ever choice of initial populations, one or the other species eventuall dies out E XAMPLE 728 (Coeistence) Set the parameters to be a 3, b 2, c 4, and d 3 The nullclines are shown in Figure 72 In this case there is a stead state at (, ) (2 3, 2) which is attracting The basin of this stead state includes 32

5 77 LOTKA-VOLTERRA M ODELS Figure 72 Competing species, coeistence The phase plane shows an attracting equilibrium in which both species survive The -nullcline 3 3 has smaller -intercept than the -nullcline According to Eercise T78, the equilibrium ( 2, ) is asmptoticall stable All 3 2 initial conditions with 0and 0 are in the basin of this equilibrium the entire first quadrant, as shown in Figure 72 Ever set of nonzero starting populations moves toward this equilibrium of coeisting populations E XERCISE T78 Consider the general competing species equation (747) with positive parameters a, b, c, d Show that there is an equilibrium with both populations positive if and onl if either (i) both a b and c d are greater than one, or (ii) both a b and c d are less than one Show that a positive equilibrium in is asmptoticall stable if and onl if the -intercept of the -nullcline is less than the -intercept of the -nullcline E XAMPLE 729 (Predator-Pre) We eamine a different interaction between species in this eample in which one population is pre to the other A simple model of this interaction is given b the following equations: ẋ a b where a, b, c, andd are positive constants ẏ c d (748) 33

6 D IFFERENTIAL E QUATIONS E XERCISE T79 Eplain the contribution of each term, positive or negative, to the predatorpre model (748) Sstem (748) has two equilibria, (0, 0) and ( c d, a b ) There are also nullclines; namel, ẋ 0when 0orwhen a b, and ẏ 0when 0orwhen c d Figure 722 shows these nullclines together with an indication of the flow directions in the phase plane Unlike previous eamples, there are no trapping regions Solutions appear to ccle about the nontrivial equilibrium ( c d, a b ) Do the spiral in, spiral out, or are the periodic? First, check the eigenvalues of the Jacobian at ( c d, a b ) E XERCISE T720 Find the Jacobian Df for (748) Verif that the eigenvalues of Df( c d, a b )are pure imaginar a/b a/b c/d c/d Figure 722 Predator-pre vector field and phase plane The vector field shows equilibria at (0, 0) and ( c, a ) Nullclines are the -ais, d b the -ais, and the lines c and a There are no trapping regions The d b curves shown are level sets of the Lapunov function E SinceĖ 0, solutions starting on a level set must sta on that set The solutions travel periodicall around the level sets in the counterclockwise direction 34

7 77 LOTKA-VOLTERRA M ODELS Since the sstem is nonlinear and the eigenvalues are pure imaginar, we can conclude nothing about the stabilit of ( c d, a b ) Fortunatel, we have a Lapunov function E XERCISE T72 Let E(, ) d c ln b a ln K, where a, b, c, and d are the parameters in (748) and K is a constant Verif that Ė 0 along solutions and that E is a Lapunov function for the equilibrium ( c d, a b ) We can conclude that ( c d, a b ) is stable Solutions of (748) lie on level curves of E Since ( c d, a b ) is a relative minimum, these level curves are closed curves encircling the equilibrium See Figure 722 Therefore, solutions of this predator-pre sstem are periodic for initial conditions near ( c d, a b ) In fact, ever initial condition with and both positive lies on a periodic orbit 35

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