520 Chapter 9. Nonlinear Differential Equations and Stability. dt =

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1 5 Chapter 9. Nonlinear Differential Equations and Stabilit dt L dθ. g cos θ cos α Wh was the negative square root chosen in the last equation? (b) If T is the natural period of oscillation, derive the formula T 4 L dθ. g cos θ cos α α (c) B using the identities cos θ sin (θ/) and cos α sin (α/), followed b the change of variable sin(θ/) k sin φ with k sin(α/), show that L π/ dφ T 4. g k sin φ The integral is called the elliptic integral of the first kind. Note that the period depends on the ratio L/g and also on the initial displacement α through k sin(α/). (d) B evaluating the integral in the epression for T, obtain values for T that ou can compare with the graphical estimates ou obtained in Problem A generalization of the damped pendulum equation discussed in the tet, or a damped spring mass sstem, is the Liénard 4 equation d dt d + c() + g(). dt If c() is a constant and g() k, then this equation has the form of the linear pendulum equation [replace sin θ with θ in Eq. () of Section 9.]; otherwise, the damping force c() d/dt and the restoring force g() are nonlinear. Assume that c is continuousl differentiable, g is twice continuousl differentiable, and g(). (a) Write the Liénard equation as a sstem of two first order equations b introducing the variable d/dt. (b) Show that (, ) is a critical point and that the sstem is locall linear in the neighborhood of (, ). (c) Show that if c() > and g () >, then the critical point is asmptoticall stable, and that if c() <org () <, then the critical point is unstable. Hint: Use Talor series to approimate c and g in the neighborhood of. 9.4 Competing Species In this section and the net, we eplore the application of phase plane analsis to some problems in population dnamics. These problems involve two interacting populations and are etensions of those discussed in Section.5, which dealt with a single population. Although the equations discussed here are etremel simple 4 Alfred-Marie Liénard ( ), professor at l École des Mines in Paris, worked in electricit, mechanics, and applied mathematics. The results of his investigation of this differential equation were published in 98.

2 9.4 Competing Species 5 compared to the ver comple relationships that eist in nature, it is still possible to acquire some insight into ecological principles from a stud of these model problems. The same, or similar, models have also been used to stud other tpes of competitive situations, for instance, businesses competing in the same market. Suppose that in some closed environment there are two similar species competing for a limited food suppl for eample, two species of fish in a pond that do not pre on each other but do compete for the available food. Let and be the populations of the two species at time t. As discussed in Section.5, we assume that the population of each of the species, in the absence of the other, is governed b a logistic equation. Thus d/dt (ɛ σ ), (a) d/dt (ɛ σ ), (b) respectivel, where ɛ and ɛ are the growth rates of the two populations, and ɛ /σ and ɛ /σ are their saturation levels. However, when both species are present, each will tend to diminish the available food suppl for the other. In effect, the reduce each other s growth rates and saturation populations. The simplest epression for reducing the growth rate of species due to the presence of species is to replace the growth rate factor ɛ σ in Eq. (a) b ɛ σ α, where α is a measure of the degree to which species interferes with species. Similarl, in Eq. (b) we replace ɛ σ b ɛ σ α. Thus we have the sstem of equations d/dt (ɛ σ α ), d/dt (ɛ σ α ). The values of the positive constants ɛ, σ, α, ɛ, σ, and α depend on the particular species under consideration and in general must be determined from observations. We are interested in solutions of Eqs. () for which and are nonnegative. In the following two eamples we discuss two tpical problems in some detail. At the end of the section we return to the general equations (). () EXAMPLE Discuss the qualitative behavior of solutions of the sstem d/dt ( ), d/dt (.75.5). We find the critical points b solving the sstem of algebraic equations ( ), (.75.5). (4) The first equation can be satisfied b choosing ; then the second equation requires that or.75. Similarl, the second equation can be satisfied b choosing, and then the first equation requires that or. Thus we have found three critical points, namel, (, ), (,.75), and (, ). If neither nor is zero, then Eqs. (4) are also satisfied b solutions of the sstem,.75.5, (5) which leads to a fourth critical point (.5,.5). These four critical points correspond to equilibrium solutions of the sstem (3). The first three of these points involve the etinction of one or both species; onl the last corresponds to the long-term survival of both species. Other solutions are represented as curves or trajectories in the -plane that describe the evolution (3)

3 5 Chapter 9. Nonlinear Differential Equations and Stabilit of the populations in time. To begin to discover their qualitative behavior, we can proceed in the following wa. First observe that the coordinate aes are themselves trajectories. This follows directl from Eqs. (3) since d/dt on the -ais (where ) and, similarl, d/dt on the -ais (where ). Thus no other trajectories can cross the coordinate aes. For a population problem onl nonnegative values of and are significant, and we conclude that an trajector that starts in the first quadrant remains there for all time. A direction field for the sstem (3) in the positive quadrant is shown in Figure 9.4.; the black dots in this figure are the critical points or equilibrium solutions. Based on the direction field, it appears that the point (.5,.5) attracts other solutions and is therefore asmptoticall stable, while the other three critical points are unstable. To confirm these conclusions, we can look at the linear approimations near each critical point FIGURE 9.4. Critical points and direction field for the sstem (3). The sstem (3) is locall linear in the neighborhood of each critical point. There are two was to obtain the linear sstem near a critical point (X, Y). First, we can use the substitution X + u, Y + v in Eqs. (3), retaining onl the terms that are linear in u and v. Alternativel, we can evaluate the Jacobian matri J at each critical point to obtain the coefficient matri in the approimating linear sstem; see Eq. (3) in Section 9.3. When several critical points are to be investigated, it is usuall better to use the Jacobian matri. For the sstem (3) we have F(, ) ( ), G(, ) (.75.5), (6) so ( J ). (7) We will now eamine each critical point in turn.

4 9.4 Competing Species 53,. This critical point corresponds to a state in which both species die as a result of their competition. B setting in Eq. (7),we see that near the origin the corresponding linear sstem is d dt (.75 The eigenvalues and eigenvectors of the sstem (8) are r, ξ () ; r.75, ξ () ). (8), (9) so the general solution of the sstem is c e t + c e.75t. () Thus the origin is an unstable node of both the linear sstem (8) and the nonlinear sstem (3). In the neighborhood of the origin, all trajectories are tangent to the -ais ecept for one trajector that lies along the -ais.,. This corresponds to a state in which species survives the competition but species does not. B evaluating J from Eq. (7) at (, ), we find that the corresponding linear sstem is d dt Its eigenvalues and eigenvectors are r, ξ () ( u v.5 ; r.5, ξ () ) u. () v 4, () 5 and its general solution is u 4 c e t + c e.5t. (3) v 5 Since the eigenvalues have opposite signs, the point (, ) is a saddle point, and hence is an unstable equilibrium point of the linear sstem () and of the nonlinear sstem (3). The behavior of the trajectories near (, ) can be seen from Eq. (3). If c, then there is one pair of trajectories that approaches the critical point along the -ais. All other trajectories depart from the neighborhood of (, ). As t, one trajector approaches the saddle point tangent to the eigenvector ξ () whose slope is.5.,.75. In this case species survives but does not. The analsis is similar to that for the point (, ). The corresponding linear sstem is d u.5 u. (4) dt v v The eigenvalues and eigenvectors are 8 r.5, ξ () ; r.75, ξ () 3, (5)

5 54 Chapter 9. Nonlinear Differential Equations and Stabilit so the general solution of Eq. (4) is u 8 c e.5t + c e.75t. (6) v 3 Thus the point (,.75) is also a saddle point. All trajectories leave the neighborhood of this point ecept one pair that approaches along the -ais. The trajector that approaches the saddle point as t is tangent to the line with slope.375 determined b the eigenvector ξ ()..5,.5. This critical point corresponds to a mied equilibrium state, or coeistence, in the competition between the two species. The eigenvalues and eigenvectors of the corresponding linear sstem are d dt u v ( ) u v r ( + )/4.46, ( ) ξ () ; r ( )/4.854, ( ) ξ (). Therefore the general solution of Eq. (7) is ( ) ( u c e.46t + c v ) (7) (8) e.854t. (9) Since both eigenvalues are negative, the critical point (.5,.5) is an asmptoticall stable node of the linear sstem (7) and of the nonlinear sstem (3). All trajectories approach the critical point as t. One pair of trajectories approaches the critical point along the line with slope FIGURE 9.4. A phase portrait of the sstem (3).

6 9.4 Competing Species 55 / determined from the eigenvector ξ (). All other trajectories approach the critical point tangent to the line with slope / determined from the eigenvector ξ (). A phase portrait for the sstem (3) is shown in Figure B looking closel at the trajectories near each critical point, ou can see that the behave in the manner predicted b the linear sstem near that point. In addition, note that the quadratic terms on the right side of Eqs. (3) are all negative. Since for and large and positive these terms are the dominant ones, it follows that far from the origin in the first quadrant both and are negative; that is, the trajectories are directed inward. Thus all trajectories that start at a point (, ) with > and > eventuall approach the point (.5,.5). EXAMPLE Discuss the qualitative behavior of the solutions of the sstem d/dt ( ), d/dt ( ), () when and are nonnegative. Observe that this sstem is also a special case of the sstem () for two competing species. Once again, there are four critical points, namel, (, ), (, ), (, ), and (.5,.5), corresponding to equilibrium solutions of the sstem (). Figure shows a direction field for the sstem (), together with the four critical points. From the direction field it appears that the mied equilibrium solution (.5,.5) is a saddle point, and therefore unstable, while the FIGURE Critical points and direction field for the sstem ().

7 56 Chapter 9. Nonlinear Differential Equations and Stabilit points (, ) and (, ) are asmptoticall stable. Thus, for competition described b Eqs. (), one species will eventuall overwhelm the other and drive it to etinction. The surviving species is determined b the initial state of the sstem. To confirm these conclusions, we can look at the linear approimations near each critical point. For later use we record the Jacobian matri J for the sstem (): F (, ) F (, ) J. () G (, ) G (, ) ,. Using the Jacobian matri J from Eq. () evaluated at (, ), we obtain the linear sstem d, () dt.5 which is valid near the origin. The eigenvalues and eigenvectors of the sstem () are r, ξ () ; r.5, ξ (), (3) so the general solution is c e t + c e.5t. (4) Therefore the origin is an unstable node of the linear sstem () and also of the nonlinear sstem (). All trajectories leave the origin tangent to the -ais ecept for one trajector that lies along the -ais.,. The corresponding linear sstem is d dt Its eigenvalues and eigenvectors are r, ξ () ( u v.5 ; r.5, ξ () ) u. (5) v 4, (6) 3 and its general solution is u 4 c e t + c e.5t. (7) v 3 The point (, ) is an asmptoticall stable node of the linear sstem (5) and of the nonlinear sstem (). If the initial values of and are sufficientl close to (, ), then the interaction process will lead ultimatel to that state; that is, to the survival of species and the etinction of species. There is one pair of trajectories that approaches the critical point along the -ais. All other trajectories approach (, ) tangent to the line with slope 3/4 that is determined b the eigenvector ξ ().

8 9.4 Competing Species 57,. The analsis in this case is similar to that for the point (, ). The appropriate linear sstem is d u u. (8) dt v.5.5 v The eigenvalues and eigenvectors of this sstem are r, ξ () ; r.5, ξ () 3, (9) and its general solution is u c e t + c e.5t. (3) v 3 Thus the critical point (, ) is an asmptoticall stable node of both the linear sstem (8) and the nonlinear sstem (). All trajectories approach the critical point tangent to the -ais ecept for one trajector that approaches along the line with slope 3..5,.5. The corresponding linear sstem is d u.5.5 u. (3) dt v v The eigenvalues and eigenvectors are r 5 + ( ) , ξ () 6 ( 3, 57)/8.387 r ( ).7844, ξ () ( )/8,.5687 (3) so the general solution is u c e.594t + c e.7844t. (33) v Since the eigenvalues are of opposite sign, the critical point (.5,.5) is a saddle point and therefore is unstable, as we had surmised earlier. All trajectories depart from the neighborhood of the critical point ecept for one pair that approaches the saddle point as t.as the approach the critical point, the entering trajectories are tangent to the line with slope ( 57 3)/ determined from the eigenvector ξ ().There is also a pair of trajectories that approach the saddle point as t. These trajectories are tangent to the line with slope.387 corresponding to ξ () A phase portrait for the sstem () is shown in Figure Near each of the critical points the trajectories of the nonlinear sstem behave as predicted b the corresponding linear approimation. Of particular interest is the pair of trajectories that enter the saddle point. These trajectories form a separatri that divides the first quadrant into two basins of attraction. Trajectories starting above the separatri ultimatel approach the node at (, ), while trajectories starting below the separatri approach the node at (, ). If the initial state lies precisel on the separatri, then the solution (, ) will approach the saddle point as t. However, the slightest perturbation of the point (, ) as it follows this trajector will dislodge the point from the separatri and cause it to approach one of the nodes instead. Thus, in practice, one species will survive the competition and the other will not.

9 58 Chapter 9. Nonlinear Differential Equations and Stabilit.5.5 Separatri FIGURE A phase portrait of the sstem (). Eamples and show that in some cases the competition between two species leads to an equilibrium state of coeistence, while in other cases the competition results in the eventual etinction of one of the species. To understand more clearl how and wh this happens, and to learn how to predict which situation will occur, it is useful to look again at the general sstem (). There are four cases to be considered, depending on the relative orientation of the lines ɛ σ α and ɛ σ α, (34) as shown in Figure These lines are called the - and -nullclines, respectivel, because is zero on the first and is zero on the second. In each part of Figure the -nullcline is the solid line and the -nullcline is the dashed line. Let (X, Y) denote an critical point in an one of the four cases. As in Eamples and, the sstem () is locall linear in the neighborhood of this point because the right side of each differential equation is a quadratic polnomial. To stud the sstem () in the neighborhood of this critical point, we can look at the corresponding linear sstem obtained from Eq. (3) of Section 9.3 d dt ( u ɛ σ X α Y α X v α Y ɛ σ Y α X ) u. (35) v

10 9.4 Competing Species 59 / σ / α / α / σ / σ (a) / α / α / σ (b) / σ / α / α / σ FIGURE / α / σ / σ / α (c) (d) The various cases for the competing-species sstem (). The -nullcline is the solid line and the -nullcline is the dashed line. We now use Eq. (35) to determine the conditions under which the model described b Eqs. () permits the coeistence of the two species and. Of the four possible cases shown in Figure 9.4.5, coeistence is possible onl in cases (c) and (d). In these cases the nonzero values of X and Y are obtained b solving the algebraic equations (34); the result is X ɛ σ ɛ α σ σ α α, Y ɛ σ ɛ α σ σ α α. (36) Further, since ɛ σ X α Y and ɛ σ Y α X, Eq. (35) immediatel reduces to d u σ X α X u. (37) dt v α Y σ Y v The eigenvalues of the sstem (37) are found from the equation Thus r + (σ X + σ Y)r + (σ σ α α )XY. (38) r, (σ X + σ Y) ± (σ X + σ Y) 4(σ σ α α )XY. (39) If σ σ α α <, then the radicand of Eq. (39) is positive and greater than (σ X + σ Y). Thus the eigenvalues are real and of opposite sign. Consequentl,

11 53 Chapter 9. Nonlinear Differential Equations and Stabilit the critical point (X, Y) is an (unstable) saddle point, and coeistence is not possible. This is the case in Eample, where σ, α, σ.5, α.75, and σ σ α α.5. On the other hand, if σ σ α α >, then the radicand of Eq. (39) is less than (σ X + σ Y). Thus the eigenvalues are real, negative, and unequal, or comple with negative real part. A straightforward analsis of the radicand of Eq. (39) shows that the eigenvalues cannot be comple (see Problem 7). Thus the critical point is an asmptoticall stable node, and sustained coeistence is possible. This is illustrated b Eample, where σ, α, σ, α.5, and σ σ α α.5. Let us relate this result to Figures 9.4.5c and 9.4.5d. In Figure 9.4.5c we have ɛ σ > ɛ α or ɛ α >ɛ σ and ɛ σ > ɛ α or ɛ α >ɛ σ. (4) These inequalities, coupled with the condition that X and Y given b Eqs. (36) be positive, ield the inequalit σ σ <α α. Hence in this case the critical point is a saddle point. On the other hand, in Figure 9.4.5d we have ɛ σ < ɛ α or ɛ α <ɛ σ and ɛ σ < ɛ α or ɛ α <ɛ σ. (4) Now the condition that X and Y be positive ields σ σ >α α. Hence the critical point is asmptoticall stable. For this case we can also show that the other critical points (, ), (ɛ /σ,), and (, ɛ /σ ) are unstable. Thus for an positive initial values of and, the two populations approach the equilibrium state of coeistence given b Eqs. (36). Equations () provide the biological interpretation of the result that whether coeistence occurs depends on whether σ σ α α is positive or negative. The σ s are a measure of the inhibitor effect that the growth of each population has on itself, while the α s are a measure of the inhibiting effect that the growth of each population has on the other species. Thus, when σ σ >α α, interaction (competition) is weak and the species can coeist; when σ σ <α α, interaction (competition) is strong and the species cannot coeist one must die out. PROBLEMS Each of Problems through 6 can be interpreted as describing the interaction of two species with populations and. In each of these problems carr out the following steps. (a) Draw a direction field and describe how solutions seem to behave. (b) Find the critical points. (c) For each critical point find the corresponding linear sstem. Find the eigenvalues and eigenvectors of the linear sstem; classif each critical point as to tpe, and determine whether it is asmptoticall stable, stable, or unstable. (d) Sketch the trajectories in the neighborhood of each critical point. (e) Compute and plot enough trajectories of the given sstem to show clearl the behavior of the solutions. (f) Determine the limiting behavior of and as t, and interpret the results in terms of the populations of the two species.. d/dt (.5.5) d/dt (.75). d/dt (.5.5) d/dt (.5.5)

12 9.4 Competing Species d/dt (.5.5 ) d/dt (.5) 5. d/dt ( ) d/dt (.5 ) 4. d/dt (.5.5 ) d/dt (.75.5) 6. d/dt ( +.5) d/dt ( ) 7. Consider the eigenvalues given b Eq. (39) in the tet. Show that (σ X + σ Y) 4(σ σ α α )XY (σ X σ Y) + 4α α XY. Hence conclude that the eigenvalues can never be comple. 8. Two species of fish that compete with each other for food, but do not pre on each other, are bluegill and redear. Suppose that a pond is stocked with bluegill and redear, and let and be the populations of bluegill and redear, respectivel, at time t. Suppose further that the competition is modeled b the equations d/dt (ɛ σ α ), d/dt (ɛ σ α ). (a) If ɛ /α >ɛ /σ and ɛ /σ >ɛ /α, show that the onl equilibrium populations in the pond are no fish, no redear, or no bluegill. What will happen for large t? (b) If ɛ /σ >ɛ /α and ɛ /α >ɛ /σ, show that the onl equilibrium populations in the pond are no fish, no redear, or no bluegill. What will happen for large t? 9. Consider the competition between bluegill and redear mentioned in Problem 8. Suppose that ɛ /α >ɛ /σ and ɛ /α >ɛ /σ, so, as shown in the tet, there is a stable equilibrium point at which both species can coeist. It is convenient to rewrite the equations of Problem 8 in terms of the carring capacities of the pond for bluegill (B ɛ /σ ) in the absence of redear and for redear (R ɛ /σ ) in the absence of bluegill. (a) Show that the equations of Problem 8 take the form ( d dt ɛ B γ ) ( B d, dt ɛ R γ ) R, where γ α /σ and γ α /σ. Determine the coeistence equilibrium point (X, Y) in terms of B, R, γ, and γ. (b) Now suppose that a fisherman fishes onl for bluegill with the effect that B is reduced. What effect does this have on the equilibrium populations? Is it possible, b fishing, to reduce the population of bluegill to such a level that the will die out?. Consider the sstem () in the tet, and assume that σ σ α α. (a) Find all the critical points of the sstem. Observe that the result depends on whether σ ɛ α ɛ is zero. (b) If σ ɛ α ɛ >, classif each critical point and determine whether it is asmptoticall stable, stable, or unstable. Note that Problem 5 is of this tpe. Then do the same if σ ɛ α ɛ <. (c) Analze the nature of the trajectories when σ ɛ α ɛ.. Consider the sstem (3) in Eample of the tet. Recall that this sstem has an asmptoticall stable critical point at (.5,.5), corresponding to the stable coeistence of the two population species. Now suppose that immigration or emigration occurs at the constant rates of δa and δb for the species and, respectivel. In this case Eqs. (3) are replaced b d/dt ( ) + δa, d/dt (.75.5) + δb. (i) The question is what effect this has on the location of the stable equilibrium point.

13 53 Chapter 9. Nonlinear Differential Equations and Stabilit (a) To find the new critical point, we must solve the equations ( ) + δa, (.75.5) + δb. (ii) One wa to proceed is to assume that and are given b power series in the parameter δ; thus + δ +, + δ +. (iii) Substitute Eqs. (iii) into Eqs. (ii) and collect terms according to powers of δ. (b) From the constant terms (the terms not involving δ), show that.5 and.5, thus confirming that, in the absence of immigration or emigration, the critical point is (.5,.5). (c) From the terms that are linear in δ, show that 4a 4b, a + 4b. (iv) (d) Suppose that a > and b > so that immigration occurs for both species. Show that the resulting equilibrium solution ma represent an increase in both populations, or an increase in one but a decrease in the other. Eplain intuitivel wh this is a reasonable result.. The sstem, γ (.5)( ) results from an approimation to the Hodgkin Hule 5 equations, which model the transmission of neural impulses along an aon. (a) Find the critical points and classif them b investigating the approimate linear sstem near each one. (b) Draw phase portraits for γ.8 and for γ.5. (c) Consider the trajector that leaves the critical point (, ). Find the value of γ for which this trajector ultimatel approaches the origin as t. Draw a phase portrait for this value of γ. Bifurcation Points. Consider the sstem where α is a parameter. The equations F(,, α), G(,, α), (i) F(,, α), G(,, α) (ii) determine the - and -nullclines, respectivel; an point where an -nullcline and a -nullcline intersect is a critical point. As α varies and the configuration of the nullclines changes, it ma well happen that, at a certain value of α, two critical points coalesce into one. For further variation in α, the critical point ma once again separate into two critical points, or it ma disappear altogether. Or the process ma occur in reverse: For a certain value of α, two formerl nonintersecting nullclines ma come together, creating a critical point, which, for further changes in α, ma split into two. A value of α at which such phenomena occur is a bifurcation point. It is also common for a critical point to eperience a change in its tpe and stabilit properties at a bifurcation point. Thus both the number and kind of critical points 5 Alan L. Hodgkin (94 998) and Andrew F. Hule (97 ) were awarded the Nobel Prize in phsiolog and medicine in 963 for their work on the ecitation and transmission of neural impulses. This work was done at Cambridge Universit; its results were first published in 95.

14 9.5 Predator Pre Equations 533 ma change abruptl as α passes through a bifurcation point. Since a phase portrait of a sstem is ver dependent on the location and nature of the critical points, an understanding of bifurcations is essential to an understanding of the global behavior of the sstem s solutions. In each of Problems 3 through 6: (a) Sketch the nullclines and describe how the critical points move as α increases. (b) Find the critical points. (c) Let α. Classif each critical point b investigating the corresponding approimate linear sstem. Draw a phase portrait in a rectangle containing the critical points. (d) Find the bifurcation point α at which the critical points coincide. Locate this critical point and find the eigenvalues of the approimate linear sstem. Draw a phase portrait. (e) For α>α there are no critical points. Choose such a value of α and draw a phase portrait , 3 α 4. 3 α, , α + 6. α +, Problems 7 through 9 deal with competitive sstems much like those in Eamples and, ecept that some coefficients depend on a parameter α. In each of these problems assume that,, and α are alwas nonnegative. In each of Problems 7 through 9: (a) Sketch the nullclines in the first quadrant, as in Figure For different ranges of α our sketch ma resemble different parts of Figure (b) Find the critical points. (c) Determine the bifurcation points. (d) Find the Jacobian matri J and evaluate it for each of the critical points. (e) Determine the tpe and stabilit propert of each critical point. Pa particular attention to what happens as α passes through a bifurcation point (f) Draw phase portraits for the sstem for selected values of α to confirm our conclusions. 7. d/dt ( ), d/dt (α.5) 8. d/dt ( ), d/dt (.75 α.5) 9. d/dt ( ), d/dt [α (α )] 9.5 Predator Pre Equations In the preceding section we discussed a model of two species that interact b competing for a common food suppl or other natural resource. In this section we investigate the situation in which one species (the predator) pres on the other species (the pre), while the pre lives on a different source of food. For eample, consider foes and rabbits in a closed forest: The foes pre on the rabbits, the rabbits live on the vegetation in the forest. Other eamples are bass in a lake as predators and redear as pre, or ladbugs as predators and aphids as pre. We emphasize again that a model involving onl two species cannot full describe the comple relationships among species that actuall occur in nature. Nevertheless, the stud of simple models is the first step toward an understanding of more complicated phenomena.