persistence of predator-prey systems in an uncertain environment

J. Math. Biology 10, 65-77 (1989) Journal of MatheMatical

Wologg O by Springer-Verlag 1980

Persistence of Predator-Prey Systems in an Uncertain Environment

Gary W. Harrison

Department of Mathematics, University of Georgia, Athens, GA 30602, USA

Summary. The time derivatives of prey and predator populations are assumed to satisfy a set of inequalities, instead of a precise differential equation, reflecting an uncertain environmental and/or lack of knowledge by the modeler. A system of differential equations is found whose solution gives the boundary of a persistent set, which is positive flow invariant for any system satisfying the inequalities. Conditions are given for the persistent set to be bounded away from both axes, which show that resonance effects cannot drive either predator or prey to extinction if that does not happen for an autonomous system satisfying the inequalities. In general predator-prey systems are more persistent when there is strong asymptotic stability, when there is correlation between prey and predator dynamics, when the effect of perturbations is density dependent, and are more persistent under perturbations of the prey than of the predator.

Key words: Preda to r -p rey- Persistence - Stabi l i ty- Differential inequalities.

1. Introduction

Assume that the dynamics of a prey population x(t) and its predator y(t) are not known exactly but are known to satisfy the inequalities

f (x , y) ~ 2 <~ f(x , y)

g(x, y) <. ~ <. 9(x, y) (1)

where" -- d/dt. The predator-prey nature of the interaction leads to the following assumptions (subscripts indicate partial derivatives):

1.~ <O,]~ <O, gx>O, Ox>O. 2. f (0 , y) = f (0 , y) = 0 and _g(x, 0) = g(x, 0) = 0. 3. There is a finite R > 0 such that f(K,O)=O, f (x , 0 ) < 0 for x > K ,

fx(R, O) < O. 4. There are functions 7 and ~ such that g(x, y) = yT(x, y) and 0(x, y) = y~(x, y).

~y(x,y) <~ O, and there a re positive values ]21 and L2 such that ~(L1,0)= 0 and 7(L2, 0) = 0.

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66 G.W. Harrison

Assumption 3 says that the prey population is itself limited by its resources so that it cannot grow above a maximum level. Assumptions 1 and 4 imply that the predator population will not grow if the prey level is below La but that a small predator population will always grow if the prey level is above L2.

The uncertainty in the growth rates 2 and )~ satisfying (1) can reflect the problems of the species trying to survive in a natural environment or of the scientist trying to model their population dynamics. There is enough uncertainty in the scientist's knowledge of the factors affecting future growth rates that he cannot completely predict them. Likewise there is enough variability in the environment that the species cannot "know", that is, be preadapted, to all possible situations affecting its growth. There must be strong mechanisms controlling the growth rates of the species to enable them to persist in spite of this uncertainty, and the scientist's job is to find these fundamental factors governing the population dynamics and show that they are strong enough to override the secondary factors that he has not included.

One way that a modeler could arrive at the inequalities (1) is to start with a pair of differential equations such as

2 = f ( x , y ) = r x ( 1 - K ) - b l ( 1 - e - a ~ X ) y (2)

= O(x,y) = y(b2(1 - e -d2x) - p),

which were used by Caughley (1976) with r = 0.8, K = 3000, bl = 1.2, dl = 0.001, b2 = 1.5, d2 -- 0.001, and p -- 1.1 to model a herbivore-vegetation system, and recognize that the parameters are not completely known and may in fact vary over time as the environment changes, i.e. r = r(t), K = K(t), etc. If one assumes interval bounds for the parameters, i.e. _r < r = r(t) < L _K < K = K(t) < K', etc. thenf , f , g, 9 can be defined by

f ( x , y ) = max f ( x , y ) r,K,b2,dl

f (x ,y ) = min f ( x , y ) r,K,bl,dt

(3.1)

(3.2)

(3.3)

(3.4)

O(x, y) = max g(x, y) b2,d2,p

g(x, y) = min 9(x, y). b2,d2,11

Thus I do not assume that x(t) and y(t) satisfy autonomous differential equations, only that the bounding functions are autonomous.

Since exact trajectories cannot be computed without more precise knowledge of 2 and)), the goal will be the find sets which are persistent according to the following definition:

Definition. A set P c R 2 is persistent for the system (1) if every trajectory (x(t), y(t)) which satisfies the inequalities (1) and has (x(to), y(to)) in P for some time to remains in P for all t > to.

In other words, a set P is persistent if it is positive flow invariant for the entire family of systems that satisfy the inequalities (1). Persistence is the most important

Persistence of Predator-Prey Systems 67

type of stability for ecological systems, since the only way both species x and y can survive in an uncertain environment is for there to be a persistent set in the interior of the positive quadrant, R2+. (See Harrison, 1979a.) Recent investigations by Freedman and Waltman (1977) and Gard and Hallam (1979) aim to find conditions for persistence, in the sense that trajectories do not approach the boundary of R 2 in either finite time or as t ~ o% when the dynamics o fx and y are exactly known. The goal here is to find conditions for persistence in spite of the dynamics varying randomly over time, and it is hoped that the persistent sets found will be far enough away from the boundary of R 2 to keep the species off the endangered species list. (Also note that differential equation models are not really appropriate if x or y become too small.)

2. Theory

If the system (1) were quasimonotone (2' > 0,s > 0, 9x > 0, 9x > 0 - mutualistic interaction) or mixed quasimonotone (fy < 0,~. < 0, 0x < 0, 9x < 0-competi t ive interaction) then standard multidimensional comparison theorems could be used to obtain a precise envelope (i.e. bounds such as _x(t) ~< x(t) <<. ~(t)) for the family of trajectories satisfying (1) (Lakshmikantham and Leela, 1969; Walter, 1970; Harrison, 1977). But according to assumption t the system is definitely not (mixed) quasimonotone. The oscillatory nature of the trajectories of predator-prey systems makes it difficult to obtain an envelope over time for the trajectories satisfying (1), due to what Moore (1966) calls the "wrapping effect". (See also Harrison, 1977, 1979b.) The following lemma gives us a way to approach this problem.

L e m m a 1. A trajectory (x(t),y(t)) cannot cross the trajectory (x*(t),y*(t)) in the direction of the outward normal (.9", - 2*) at any point where

22" > 0 and ~ . <_x (4.1)

o r

2" 2 - - (4 .2 ) f;~* > O and f;. > ~ .

Proof. The trajectory (x(t),y(t)) cannot cross x*(t),y*(t) if the angle between the tangent (2,~) and the outward normal ( ~ * , - 2*) is greater than 90 ~ i.e. if 23~* - 3~2" < 0. Dividing by 22* or j~3~* as is appropriate yields (3).

L e m m a 2. Under assumptions 1 - 4 the isocline f = 0 lies above the isoeline f = 0 and the isocline 0 = 0 lies to the left o f the isoeline 9 = O. All four isoclines meet [-he x axis, but the isoclines j = 0 and 9 = 0 remain to the right of the line x = L1. (See Fig. 1.)

Note, however, that assumptions 1 - 4 do not guarantee that L 1 or L2 is less than * or that either f - - 0 or f = 0 meets the y axis.

T h e o r e m 1. Let (x*(t), y*( t) ) be a trajectory whose time derivatives 2 ,.9 are defined in each of four regions as follows. (See Fig. 1.)

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Reg. 2. / I Reg. 1. ~*=f,o / ~ ~*=~,o ~'* 4,o / I ';'* :~ >o

Reg. 3 . "",,, .

) K * = f : ' 0 _ " ' . "~'0 ~(*=cJ<O Reg. 4. ~"~,<'0 2:0"<:\

?*=g_>O x, ",, \

Ll L 2 ~,

Fig. 1. The solid lines are the parts of the isoclines f = 0, _f = 0, ~ = 0, and 9 = 0 that form the boundaries of regions 1 -4, defined by system (5); the rest of each isocline is indicated by a dashed line. In each region the functions used to define 2* and .9* in system (5) are indicated. The shaded area is the indeterminate region where f < 0 <J~ -9 < 0 < 0

Region 1 : I f f < 0 and O >~ 0 then

2* = f ( x * , y * ) , 9* = O(x*,y*). (5.1)

Region 2 : I f 9 < 0 and f <~ 0 then

2" = f (x*, y*), J)* = O(x*, y*). (5.2)

Region 3: I f f > 0 and g <~ 0 then

2" = f (x* ,y*) , .9* = g(x*,y*). (5.3)

Region 4: I f g > 0 and f > 0 then

2" = f (x* , y*), .9* = g(x*, y*). (5.4)

Then any trajectory satisfying the inequalities (1) cannot cross the trajectory (x*(t),y*(t)) in the direction of the outward normal (j)*, - 2*).

Pro@ Suppose a trajectory (x(t),y(t)) satisfying (1) were to cross a trajectory (x*(t), y*(t)) satisfying (5) at a point x = x*, y = y* in the direction o f the outward normal. In each region either 2 and 2" or .9 and 3~* would have the same sign. For example, in region 1 f ( x , y ) < 2 < T(x,y) = 2* < 0. Hence either (4.1) or (4.2) of Lemma 1 holds and the trajectory (x(t),y(t)) cannot cross the trajectory (x*( t), y*( t) ).

Remarks. There is an indeterminate region I bounded by the four isoclines shown in Fig. 1 where f lx , y) <~ 0 4 f ( x , y ) and 9(x,y) <<_ 0 <~ O(x,y). Here it is impossible to define 2* and-j)* in such a way that either 2 and 2" or )~ and )~* must have the same sign, and Lemma 1 is inapplicable. In fact, at a point in I t h e inequalities (1) allow both 2 and.9 to be either positive or negative, and hence a trajectory th rough a point in ! could go in any direction. The indeterminate region I is the set o f all the equilibrium points o f all the au tonomous systems satisfying (1), and it will be shown later that its size may be one good measure of the overall stability (in a biological sense) o f the system. Also note that the directions o f the trajectories are such that no solution o f (5) can enter I f rom outside.

Figure 2 shows a solution o f the system (5) when f , f , 9, 9 are computed for system (2) according to (3) with 2800 ~< K = K(t) <~ 3200, 1.0 < p = p(t) < 1.2,


Fig. 2. A persistent set for equation (2) when 2800 ~ K ~< 3200, 1.0 ~< p ~< 1.2 and the other parameters are as given in the text. The smooth curve is a solution of (5) that spirals in toward a closed curve that is a stable limit cycle of (5) and forms the boundary of the persistent set. No trajectory satisfying (1) can cross it from the inside to the outside. The irregular curve, corn- puted from (11), is an example of a typical trajectory satisfying (1)

1000

500

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I I 1000 2000 3 0 0 0

and all other parameters constant . Since no other solution o f (1) with coefficients satisfying these condit ions can cross the spiral f rom the inside to the outside, the inside o f the spiral is a persistent set for this system.

3. Properties of System (5)

Since solutions o f system (5) are used to determine the existence and boundaries o f persistent sets and whether any trajectories satisfying (1) become unbounded or approach x = 0 or y = 0, it is impor tan t to unders tand the properties o f the solutions o f (5).

System (5) is obviously a differential equat ion with discontinuous r ight-hand side. The reader who is interested in a thorough discussion of such systems is referred to Fil ippov (1964, especially Sect. 7). Normal ly , however, the bounding functionsf,~, g and ~ are such that for each of the separate systems (5.1) - (5.4) there will exist aun~que solution th rough any point in the closure of region 1 to region 4, respectively. In this case one does not need such an extensive theory as that developed by Fil ippov to see that there is a unique solution o f the entire system (5) th rough any initial point formed by joining the appropria te unique solutions o f the separate subsystems.

Lemma 3. Under assumptions 1 - 4 no solution of system (5) can become unbounded.

Proof. Consider a solution o f (5) (x*(t), y*(t)) starting at a time to at a point (K, Y0) in region 1, with Yo > Ym = m a x i m um of y on the isocline f ( x , y ) = 0 for L1 ~ x ~< K. F r o m here x*(t) decreases and y*(t) increases. By assumption 1 ~*<~ maxLl<x<Kf(x, yo) and the isocline ~(x ,y )= 0 is to the right o f the line x = LI , so that the trajectory enters region 2 at a finite time tl. Also by assumptions 1 and 4 ~* ~< [g(K, yo)/yo]y*, so that y*(tl) <~ exp[(~(K, yo)/Yo)(tl - to)]. Hence the trajectory (x*(t), y*(t)) enters region 2 at a point (~,y~) with Ym < Y < ~ . (See Figure 3a.)

In region 2 both x*(t) and y*(t) must decrease, in region 3 x*(t) is bounded and y*(t) must decrease, and region 4 is bounded. Hence any solution o f (5) th rough a point in any of these regions must remain bounded (approach an equilibrium point,

70 G . W . Harrison

Yo)

L1 L 2 __. K

b.

~=o/ ,

Z ,,~_: o �9 �9 ,,,,,,~ .~

/ . j T ' # ' ~ .~

i , ! !

-,J - L~ LZ K K LI KLz K X

Fig. 3, Three cases depending on where the isocl inef = 0 meets the x axis. (3a) It meets the axis only at _K > L2: all trajectories satisfying (1) are bounded avTay from both axes. (3b) The isocline meets the x axis at another point J < _K: trajectories satisfying (1) to the left of the separatrix (dashed-dotted line) may approach (0, 0); those to the right are bounded away from the axis. (30 _K < L2: the solution of (5) approaches (_K, 0) so that trajectories satisfying (1) may approach the x axis

say) or enter region 1 at a point (Xl, Yl) with xl ~< Kand Yl < Yo. But then it cannot cross the solution through (K, Yo) discussed in the preceding paragraph and hence stays in the set x*(t) <~ K, y*(t) <~ ~. If a trajectory starts at a point with x* > K or Y* > Yo, then by the same argument as the preceding paragraph it enters region 2 in finite time at a finite value of y*, and must remain bounded after that.

Corollary 1. Under assumptions 1 - 4 no trajectory satisfying the inequalities (1) can become unbounded.

The reader interested in more general conditions for boundedness of predator- prey systems should consult Brauer (1979).

Under assumptions 1 - 4, the possibility of a trajectory satisfying inequalities (1) and approaching either the x or y axis (i.e. either the prey or predator becoming extinct) is determined by the nature of the isoclinef(x, y) = 0. Assumption 3 and Lemma 2 imply that there is a value _K ~< R withf(_K, 0) = O,f(x, 0) < 0 for x > _K. (See Fig. 1.)

Case 1: If f (x, 0) > 0 for all 0 < x ~< L2, it follows that _K > L2 and the isocline f ( x , y ) = 0qs above the x axis for 0 < x < L2 (see Fig. 3a), that is, there are no equilibrium points of (5) on the x axis for 0 < x ~< L2. Except for the origin, there are no equilibrium points of (5) on the y axis. But neither the origin nor any equilibrium point on the x axis with x > L2 (such as (K, 0)) can be approached by a trajectory from the interior of R 2, since ;t > 0 near the former and p > 0 near the latter; they must be saddle points. Hence any solution at a point in regions 2, 3, or 4 must pass to the next region in finite time and remains at positive distance from both axes. It enters region 1 at a point (x l , y l ) with xl < K, Yl < Yo,Yo as defined in the proof of Lemma 3, and thereafter is bounded by the solution of (5) through (K, yo). This latter trajectory cannot cross itself or the line x = K, so that all solutions of (5) are bounded away from both the x and y axes. This eliminates the possibility of a trajectory spiraling outward in such a way that it contains a sequence of points approaching an axis asymptotically.

I f f (x* , 0) = 0 for some 0 < x* ~< L2, then any trajectory of (5) passing through a point below the isoclinef(x, y) = 0 and left of x* cannot cross the isocline from


below to above and leave that region, since p must be negative. It must approach an equilibrium on the x axis. The two most likely cases where that occurs are discussed next, although they are not a complete listing of the possibilities.

Case 2: If there is a J < L 1 with f ( x , O ) < O on O < x < J , f (x , 0 ) > 0 on J < x < L2, then (J, 0) is a saddle point and (0, 0) is a stable equilibrium (see Fig. 3b). The separatrix running into the point (J, 0) divides the plane into an area where solutions of (5) (and hence all trajectories satisfying (1)) are bounded away from the axes and an area where solutions of (5) go to (0, 0), (and hence some trajectories satisfying (1) go to (0, 0)). Only, however, i ff(x, 0) = 0 for some x < J < _Kis there an area where all trajectories satisfying (1) must approach (0, 0). Biologically Case 2 means that under some conditions the population growth rate would become zero or negative at low population levels (an Allee effect), even in the absence of predation.

Case 3: If K < L2, then solutions of (5) cannot leave region 3, but must approach (_K, 0). (See Fig. 3c.) Biologically this means that the carrying capacity of the prey may sometimes be below the level required for the predator population to grow. Many trajectories satisfying (1) would still not approach the x axis, unless g7 < Li (i.e. the carrying capacity of the prey is always below the level for the predator population to grow).

Case 1 implies all trajectories satisfying (1) are bounded away from the x axis since they cannot cross solutions of (5). Hence we have the following:

Corollary 2. Under assumptions 1 - 4 , i f f ( x , 0 ) > 0 for all x <<. Lz, then all trajectories satisfying (1) are bounded away f rom the x and y axes.

If the hypothesis of Corollary 2 is violated then the system 2 = f, 3~ = 9 has solutions approaching the x axis. Hence

Corollary 3. A nonautonomous system satisfying (1) and assumptions 1 - 4 cannot have a trajectory with either species going to extinction unless an autonomous system (namely 2 = f , p = g) has a trajectory with a species going to extinction.

In other words, Corollary 3 says that there is a limit on the effects of resonance in a system satisfying the constraints in (1). The changes in growth rates that would have the greatest resonance effect and produce the most outward spiral are precisely those given in system (5), and system (5) has a solution approaching an axis only if one of the autonomous subsystems (5.1)-(5.4) does.

Lemma 4. There is a system o f differential equations

2 = F(x, y)

= G(x, y) (6)

with F(x ,y ) and G(x,y) defined and continuous on R 2 - I that has the same trajectories as the system (5).

Proof. Note that the derivatives 2* and 3~* defined by (5) are continuous functions of (x*, y*) except for finite jump discontinuities at the boundaries of the regions. Let Biz = {(x, y): O(x, y) = 0,j~(x, y) < 0), which is the boundary between region 1 and

72 G . W . Harrison

2, and let p(x, y) be the distance between (x, y) and B12. Pick a fixed e > 0, and at any point in region 1 with p(x, y) < e define

G(x, y) = O(x, y)F(x, y)/f(x, y). (7)

Define F(x, y) = f ( x , y) in the rest of region 1 and F(x, y) = f (x , y) in region 2. Define G(x, y) = O(x, y) in the rest of region 1 and in region 2 (except for a thin strip along the boundary with region 3). Now F(x, y) is continuous throughout regions 1 and 2 and since 0(x, y) = 0 on B12, G(x, y) is continuous also. But G(x, y)/F(x, y) is O(x, y)/f(x, y) in region 1 and O(x, y)/f(x, y) in region 2 (and zero on B12) so that dy/dx = dy*/dx* at all points and both systems have the same trajectories throughout regions 1 and 2.

The jump discontinuities at the other boundaries can be patched up in a similar manner.

Theorem 2. The positive limit set of any solution of system (5) which is bounded away from the x and y axes must either be a closed periodic orbit or else contain a point on the boundary of the indeterminate region L

Proof. It was already noted that system (5) (and hence system (6)) has a unique solution through any point in R2+ - 1, and, since F(x, y) and G(x, y) are continuous on Re+ - / , Poincar6-Bendixson theory can be applied to the trajectories of (6) (and hence (5)). Every trajectory satisfying (5) is bounded and by hypothesis is bounded away from the axes, so that if for some e > 0 the distance between a trajectory (x*(t), y*(t)) and I remains greater than e for all t, then (x*(t), y*(t)) is in a compact region that contains no critical points. Thus the Poincar6-Bendixson theorem implies that its positive limit set is a closed periodic orbit. If there is no such e, there must be a sequence of points (x*(4), y*(t,)) whose distance from I approaches zero as the sequence t, approaches infinity. This sequence must contain a convergent subsequence whose limit is a point in L

Figure 2 shows a case in which the positive limit set is a closed orbit. It is in fact a stable limit cycle approached by solutions of system (5) from both sides and gives the boundary of the smallest persistent set. Since this persistent set is approached by

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Io'oo zo'oo ' ' o ~

Fig. 4. When inequalities (1) are de- rived from (2) and (3) using 2800 ~< K ~< 3200, 1.34 ~< p ~< 1.36, solutions of (5) approach an equilibrium point on the boundary of the indeterminate region. A boundary of the persistent set is formed by adjoining the transversal T to the solutions of (5)


any solution of (5) that begins outside of it, any trajectory satisfying (1) must approach it. Figure 4 shows a case in which the positive limit set is an isolated point on the boundary of L In fact, it is the equilibrium point of system (5.3). In this case the boundary of a persistent set can be formed by adding an appropriate transversal, as shown in Fig. 4. Again all trajectories satisfying (1) not only remain in the persistent set if they start inside but approach it if they begin outside.

I have not been able to rule out the possibility that the positive limit set of a solution of (5) might be a closed curve that contains a point (or points) on the boundary of L If, for example, the equilibrium point P of (5.3) were a stable focus (instead of a stable node), then a solution of (5) might spiral around the region I, approaching closer to P each time around but always passing it.

4. Factors Affecting the Size of the Persistent Set

It is nice to know that trajectories cannot approach either axis, but from a biological point of view this is not good enough. As pointed out in the introduction, if a trajectory approaches too close to an axis then one species is practically extinct and a differential equation model is no longer appropriate; a stochastic model admitting the importance of random events to a small population would be better. But if the persistent set, whose boundary is a solution of (5), is reasonably far from both axes then there is no danger of extinction. If it is reasonably small then the population sizes cannot fluctuate too much, which is the meaning of the word "stability" as used in many ecological papers (Harrison, 1979a). I experimented with system (2), using equations (3) and various parameter intervals to define the bounding functions in order to determine what factors affect the size of the persistent set.

Fig. 5. When inequalities (l) are de- rived from (2) and (3) using 3200 ~< K ~< 3600, 0.84 ~< p ~< 0.86, any ho- mogeneous system satisfying (1) has an unstable equilibrium. The persistent set, whose boundary is a limit cycle of (5), is dangerously close to both axes

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74 G .W . Harrison

Asymptotic Stability

By far the strongest factor is the asymptotic stability properties of the system, as can be seen by comparing Figs. 4 and 5. The equilibrium point of any autonomous system satisfying the conditions used in Fig. 5 lies in the indeterminate region which is to the left of the "hump" in the prey isocline; thus it is unstable (Rosenzweig and MacArthur, 1963; Freedman, 1976). The equilibrium point of any autonomous system satisfying the conditions used in Fig. 4 lies below any points on the isocline to its left; thus it is globally asymptotically stable (Harrison, 1979c). Although in both cases there is a persistent set that is entered by all trajectories beginning outside of it, in Fig. 4 it is small and population levels would remain nearly constant, but in Fig. 5 it is so large and comes so near the axes that population fluctuations could be huge and biologically extinction may be possible.

Size of Indeterminate Region and Perturbations of Predator vs. Prey Dynamics

When comparing systems with similar asymptotic stability properties I found a strong correlation between the size of the persistent set and the size of the indeterminant region, and that replacing any parameter ~ in the predator equation by an interval [0.9~, 1.1~] gave a somewhat larger persistent sets than doing the same in the prey equation. The former observation is not surprising and helps to explain the latter. Since the indeterminate region is the set of all the equilibria of all autonomous systems satisfying (1), an estimate of the relative effect of uncertainty in a parameter ~ on its size is given by OxE/O~ and Oyg/d~. Differentiating implicitly the equations

f ( x E, yE, a) = 0 (8)

g(x ~, yE, ct) = O,

and using gy(x E, y~) = 0 because the predator isocline in (2) is vertical, one obtains

~XE/a~ = -- gJgx (9)

OyE/OOt = --f~/fy + gj~/fygx.

If ~ is a parameter in the predator equation, f , = 0 and the change in x E is proportional to 1/9~; if c~ is a parameter in the prey equation, g~ = 0 and the change in yE is proportional to 1/f r. The ratio Ifrl/Ig~l "~ 4 in system (2). In other words, a change in the predator density has a greater effect on the prey growth rate than vice versa, hence, if the predator growth rate is perturbed by an environmental stress it takes a larger change in the prey density to bring the system back in balance. Since inefficiencies in conversion of captured prey into predator biomass dictate that in general [fy[ > Igxl, I conclude that generally predator prey systems are less persistent under perturbations of the predator dynamics than of the prey dynamics.

Non-Density-Dependent Stresses

An environmental stress that can be represented by a change in one of the parameters in equation (2) has an effect that is proportional to the population sizes.


Fig. 6. When inequalities (1) are de- rived from (10) and (3) using 2800 ~< K~<3200, 1.0~<p~<1.2, u = 0 , 0~< v ~< 40, the indeterminate region is in two parts. All solutions of (5) approach y = 0 so that there is no persistent set that is bounded away from the x axis

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Although most natural stresses would seem to affect a population in such a density dependent manner (i.e. changing the per capita rather than the absolute birth and death rates) some stresses, especially those caused by man, may not. Hence consider the equations

2= r x ( 1 - K ) - b l ( 1 - e - e l X ) y - u (10)

3~ = y(b2(1 - - e -dzx ) -- p ) -- v, ~ ~ u ~ if, v ~ v ~ v.

Brauer and Soudack (1979a, b) have used terms u and v to represent constant rate harvesting and give a detailed analysis. Here they represent any uncertainty that does not go to zero as x or y goes to zero.

When fi > 0, ~ = 0, the situation is that of Case 2 discussed before Corollary 2, with the isoclines shown in Fig. 3b. There is a region where persistence is guaranteed, and a region where trajectories satisfying (1) may go to (0, 0). When ~5 > 0, ~ = 0 the isoclines are as shown in Fig. 6, and the indeterminate region is now in two parts. For the case shown in Fig. 6 all solutions of (5) meet the x axis, so that extinction is possible from any initial point, illustrating that a predator-prey system is less persistent if there are non-density-dependent disturbances of the birth and death rates.

Correlations Between Parameters in Prey and Predator Equations

So far I have considered the parameters in the equations for 2 and p to be independent. It is quite likely, however, that the values of some of the parameters are related (e.g. bt and b2 or dl and d2 in Eq. 2), so that it would be impossible to have certain combinations (such as 2 = f(x, y) and p = _9(x, y)) occurring at the same time. A system satisfying such further restrictions would still satisfy the inequalities (1) with f , f , 0, 9 defined by (3), so that its trajectories would remain within the persistent setcomputed from (5), and Corollaries 1,2, and 3 still apply. A

76 G.W. Harrison

smaller persistent set could be found by applying Lemma 1 directly, and using the restrictions on the parameters when finding minj)*/2* or max2*/p*.

5. Biological Interpretations and Conclusions

Ecologists have long talked about abiotic limiting factors determining the geographical range of a species. The inequalities (1) may be thought of as representing density-dependent biotic limiting factors on population growth. At extreme population levels death rates so far outweigh birth rates, or vice versa, that the population is bound to decrease or increase (represented byf(x, y) andf(x, y) or g(x , y) and 0(x, y) having the same sign). The rate may be faster or slower depending on the current environmental conditions but the change is sure to occur, because the essentially deterministic limiting factors are strong enough to override the uncertain environmental conditions.

But with a predator-prey system, that has natural oscillatory properties, there is the possibility that if changes in the growing conditions occur just when the population dynamics change from increasing to decreasing (or vice versa) that the resonance effect could lead to a trajectory that swung from higher highs to lower lows until a species became extinct. Theorem 1 (and Corollary 3), however, show that the maximum resonance effect occurs when conditions change so that growth rates satisfy (5). Hence a solution of (5) gives a boundary of a persistent set. No disturbance or sequence of disturbances, unless it were so catastrophic that it violated the inequalities (1), could push a trajectory across this boundary to the outside. Section 3 gives properties of the persistent set, including conditions for it to be bounded away from the axes. Section 4 discusses factors that affect the size of the persistent set: asymptotic stability, the size of the indeterminate region, whether the perturbations are in the prey or the predator dynamics, disturbances whose effects are not proportional to population sizes, and correlations between prey and predator dynamics.

Of course it must be remembered that solutions of (5) represent the very worst that could happen if a system satisfies the inequalities (1), not what is likely to happen. The derivatives have to shift from 2 = f (x ,y ) to 2 = f ( x , y ) , and from

= g ( x , y ) to ~ = 0(x,y), at just the right times to give the most outward trajectory found by solving (5), a highly unlikely event. A more typical trajectory is the irregular one shown in Fig. 2, found by numerically solving

s = ~f_(x,y) + (1 - ~ ) / ( x , y ) (11)

=/~g(x,y) + (1 -/~)O(x,y)

where c~ and fl were reselected at random from a uniform distribution on the interval (0, 1) after every z units of time. I found that solutions of (11) usually show irregular oscillations, similar to the pattern found by Jeffries (1974), and tend to move away from the boundary of the persistent set and cluster around the indeterminate region. At these population levels the limits on the birth rates balance the limits on the death rates close enough that stochastic factors dominate, causing irregular fluctuations in the population levels. The size ofz had some influence on how far the trajectories strayed from the indeterminate region, but it was not great.


T h u s t h e size o f the i n d e t e r m i n a t e r e g i o n , as wel l as t he size o f t he p e r s i s t e n t set,

is a n i n d i c a t o r o f t he c o n s t a n c y o f t he p o p u l a t i o n levels. I f o n e d o e s n o t ca re h o w

m u c h t h e p o p u l a t i o n levels f l u c t u a t e as l o n g as n e i t h e r species b e c o m e s ex t inc t , h e

s h o u l d l o o k a t h o w f a r t h e p e r s i s t e n t set is f r o m b o t h axes.

Acknowledgements. This work was influenced by conversations with Ernst Adams, Fred Brauer, Tom Gard, Tom Hallam, and George Johnson. It was supported by N.S.F. grant no. DEB78-03007.

References

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Received August 1, 1979/Revised January 9, 1980

persistence of predator-prey systems in an uncertain environment

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