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Persistence in predator-prey systems with ratio-dependent predator influence
Bulletin of Mathematical Biology Vol 55, No. 4, pp. 817-827, 1993 Printed m Great Britain
0092-8240/9356 00+ 0.00 PergamonPressLtd © 1993 Societyfor Mathematical Biology
PERSISTENCE IN PREDATOR-PREY SYSTEMS WITH RATIO-DEPENDENT PREDATOR INFLUENCE H. I. F R E E D M A N ? Department of Mathematics, Applied Mathematics Institute, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 (E.mail :[email protected] )
R. M. M A T H S E N ~ Department of Mathematics, North Dakota State University, Fargo, ND 58105-5057, U.S.A. Predator-prey models where one or more terms involve ratios of the predator and prey populations may not be valid mathematically unless it can be shown that solutions with positive initial conditions never get arbitrarily close to the axis in question, i.e. that persistence holds. By means of a transformation of variables, criteria for persistence are derived for two classes of such models, thereby leading to their validity. Although local extinction certainly is a common occurrence in nature, it cannot be modeled by systems which are ratio-dependent near the axes.
1. Introduction. Continuous models, usually in the form of differential equations, have formed a large part of the traditional mathematical ecology literature. In order to carry out the required mathematical analyses, they are assumed to be sufficiently smooth over their entire domain of definition so that solutions to initial value problems exist uniquely and are continuable for all positive time. In particular, in the case of predator-prey systems the domain must include the prey and predator axes, since one must know the dynamics of each population in the absence of the other (see, e.g. Freedman, 1980; Freedman and Waltman, 1984 and refs therein). Siinilar statements can be made with respect to competition models, mutualism models or models which incorporate combinations of these (Freedman, 1980; Freedman and Waltman, 1985). Recently, however, several manuscripts have been written, and more are likely to appear, which incorporate terms involving the ratio of prey-topredator or predator-to-prey. These terms have appeared in place of predator functional response terms or in the predator growth terms (Arditi and t Research partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. NSERC A4823. :~ Research carried out while visiting the Uni~cersity of Alberta. 817
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H . I . F R E E D M A N A N D R. M. M A T H S E N
Ginzburg, 1989; Lindstrom, preprint; Sarkar et al., 1991). The ecological reasons for choosing ratio dependence in Arditi and Ginzburg (1989) are related to scale. Wiens et al. (1986) have shown that choosing various temporal or spatial scales can lead to different outcomes. Introducing ratio dependence leads to a reduction of shared resources for the predator if more predators are present in the prey-to-predator ratio case, for example. It is not our purpose in this paper to discuss the ecological validity of such ratio-dependent models, but to note that such ratio dependence does lead to some mathematical problems and these have not been dealt with in the above mentioned manuscripts, and therefore we deal with them here. By virtue of ratio dependence the model is undefined, and hence discontinuous, on either the prey or predator axis. Hence, any analysis of solutions initiating with positive populations is valid only if one can show that such solutions "stay away" from the axes. By "stay away" we mean that there does not exist a sequence of times tending to infinity such that either the prey values or predator values tends to zero. A convenient way of expressing this is in terms of "persistence". In the real world, in a predator-prey setting, persistence of populations in a given environment or local extinction of one or more population may occur. Gause et al. (1936) showed that local extinction could indeed occur after only a few generations in paramecuim-didinium interactions in the laboratory. (However, Veilleux, 1979, in repeating these laboratory interactions was able to sustain both populations for over 50 cycles, until he terminated the experiment.) We feel, however, since the ratio-dependent models are not mathematically correct on the axes, they ought not to be utilized when extinction occurs unless the functional forms of the models are appropriately redefined near the axes. Hence, we are interested in the question "when does persistence occur in ratio-dependent models?" In such cases, the models are indeed valid throughout the positive orthant. In an ecological context, persistence may be defined mathematically as in Freedman and Waltman (1984, 1985). A population N(t) is said to persist (sometimes termed strongly persist) if N(0)> 0 implies N(t)> 0 for t > 0 and lim inft_.o~ N(t)>0. For related definitions in both ecological and abstract situations, see Butler et al. (1986), Butler and Waltman (1986) and Freedman and Moson (1990). One of the related definitions that will be of interest is uniform persistence (also known as permanence). N(t) is said to persist uniformly if N(t) persists and there exists 6 > 0 , independent of N(0)>0, such that lim inft_. 00N(t) >~6. Finally, we say that a system (uniformly) persists whenever each component (uniformly) persists. The standard techniques for showing persistence, which require the system to be defined on the axes, do not work here unless the system can be transformed into a dynamical system. This we are able to do for each of the
PREDATOR-PREY SYSTEMS
819
models of interest. By a change of variables we transform each of the given systems into a dynamical system in such a way that there is a one-to-one correspondence between the positive values of the predator and prey of the original system and the positive values of the transformed system. Hence, persistence in the transformed system implies persistence of predator and prey, i.e. it implies that the analyses in the ratio-dependent manuscripts are valid. The usual techniques for deriving persistence typically also imply uniform persistence. It is of interest to note that such is not the case here. In the cases considered in this paper, the usual techniques described in Freedman and Waltman (1984) or in Gard (1987) do not work, since we cannot prove that the transformed systems are dissipative. However, we are able to obtain persistence criteria utilizing geometrical arguments. In Section 2 we will consider the case where the ratio dependence is predatorto-prey and Section 3 where it is prey-to-predator. In the first case, the standard assumptions will give persistence. However, in the second case, additional assumptions will be required. Both cases are illustrated by examples. We conclude with a short discussion in Section 4. All proofs of our results are deferred to the appendix. 2. Predator-to-Prey Ratio Dependence. predator-prey model of the form:
In this section we consider a
xg(x)-- yp(x), x(0) = x o > 0 y =yf(y/x),
y(0) = Yo > 0;
(1)
where x(t) is the prey density, y(t) is the predator density and "= d/dt. We assume the following: (H1) g, P and f have second derivatives continuous in their arguments on the interval (0, oe). This is sufficient to guarantee that solutions to positive initial value problems exist uniquely at least for some positive ti.me. (H2') 9(0)>0; g'(x)<0; there exists K > 0 such that g(K)=0. For small values of the prey population, it will grow. However, there exists a carrying capacity of the environment beyond which the prey population cannot increase, even in the absence of predators. (H3) p(0)=0; p'(x)>0. The predator functional response is an increasing function of the prey, and, of course, in the absence of prey there can be no predation. Before stating our last hypothesis, we effect the change of variables
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H . I . F R E E D M A N A N D R. M. M A T H S E N
(2)
z =y/x. Then f ( y / x ) = f ( z ) .
(H4) f(0) > 0; f'(z) < 0; there exists L > 0 such that f ( L ) = O. z is small if y is small or x is large, i.e. few predators or many prey. In either case we would expect the predator population to grow. However, if the predator-to-prey ratio increases, it is reasonable to assume that the predator growth rate would decrease to the point where growth changes to loss (at y/x=L). Under these assumptions, our model includes the model of Lindstrom (preprint) and the submodel in Sarkar et al. (1991) representing microorganisms feeding on detritus. We now derive the system satisfied by x and z. By (1) and (2), xzf(z) = ~9= d/dt(xz) = x~ + z2 = x~ + z i g ( x ) - zp(x)]. Hence, system (1) transforms to: x [ g ( x ) - zp(x)], i = z [ f ( z ) - g(x) + zp(x)],
x(0) = x 0 > 0 z ( 0 ) = Yo > 0.
(3)
x0
Note that positive solutions of system (3) are in a one-to-one correspondence with positive solutions of system (1). However, system (3) can be extended to the axes in a smooth manner, thereby causing it to be a dynamical system over R2+, the non-negative quadrant. Further, since persistence deals only with positive solutions, persistence in system (3) implies persistence in system (1). We now note that the prey population in the transformed model continues to be limited by its resources, i.e. x(t) is bounded. In fact, lim supt_.o~ x(t)<~ K. Next we examine the existence of equilibria in system (3), i.e. those values of x and z for which 2 -- ~ = 0. Algebraically, this involves setting the right sides of system (3) equal to zero and solving. Geometrically, positive equilibria occur whenever the isoclines intersect, where the isoclines are the curves given by: r l :z = g(x)/p(x)
F 2 :f(z) + zp(x) = O(x).
(4)
We now have the following results on the existence of equilibria. THEOREM 1. System (3) has at most four equilibria as follows: (1) E0(0, 0) always exists. (2) E 1(K, 0) always exists. (3) E2(0, ~), where f(~)=g(0) and 0 < ~ < L , exists if and only/ff(0)>g(0).
PREDATOR-PREY SYSTEMS
(4) E*(x*, L), where g(x*)=Lp(x*) and 0 < x * < K , unique.
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always exists and is
In order to obtain our next result, which is p a r a m o u n t in proving persistence for system (3), we require some information about F 2. I f f ( 0 ) > g ( 0 ) , then (0, ~) e F 2 . If f(0) = g(0), then (0, 0) e F 2 . If f(0) < g(0), there exists a point of the form ()~, 0 ) ~ F 2, where g ( Y ) = f ( 0 ) > 0 and hence 0 < X < K . N o w consider the case where z = L. Since using equation (4) the only value of x that satisfies - - g ( x ) + L p ( x ) = O is x = x * , F 2 can cross z = L only at E*. Hence, the curve F2, once it rises above L, cannot come below it again. THEOREM 2. Let f(z) + zp(K) = 0 have at most a finite number of roots. Then z(t) is bounded for t >~O. We can now state our main result of this section. THEOREM 3. Let hypotheses (H1)-(H4) hold. Further, assume that f(z)+ zp(K) = 0 has at most a finite number of positive roots. Then system (3), and hence system (1), exhibits persistence. We now illustrate the above results by means of an example. We consider the system: 2=x(a-bx
))=y(d-~),
c~x)'
x(0)=x°>0
y(0)=yo>0.
(5)
Here, g(x) = x ( a - bx), p(x) = cx/a + x, f ( z ) = d - ez, K = a/b and L = die. We note that hypotheses (H1)-(H4) are all satisfied. We now observe that the functionf(z) + zp(K) = (ac - ae-- abe/a + ab )z + d has at most one positive zero. Hence, by T h e o r e m 4 all solutions of (5) persist and therefore any analysis of positive solutions is valid.
3. Prey-to-Predator Ratio Dependence. I n
this section we consider a
p r e d a t o r - p r e y system of the form: x(0) = Xo > 0
y(0) = Yo > 0; where 7, c > 0.
(6)
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H . I . F R E E D M A N A N D R. M. M A T H S E N
Here, we let u = x / y and we assume (H1)-(H3) hold, replacing z by u in p. This is precisely the model considered by Arditi and Ginzburg (1989). We n o w derive the transformed system for the variables x and u: fi = (y2 - x~ )/y 2 = (xg(x) - yp(u) + 7x - cxp(u) )/y = u g ( u ) - p(u) + 7 u - cup(u).
If we set:
q(u)=
p(u) u'
u>0
(7)
p'(0), u=0; then our model becomes: 2 = x[g(x) -- q(u)],
x(0) = x o > 0
fi = u[g(x) + ~ -- cp(u)-- q(u)],
u(0) = x° > 0. Yo
(8)
We note that for system (6) O < < . x ( t ) ~ m ( x o ) = m a x { x o, K}. F o r system (6) there are at most four non-negative equilibria: Eo(0, 0), /~(2, 0), /2(0, ~7) and E * ( x * , u*). F o r /~ to exist 2 > 0 must exist such that g(2) = p'(0). This occurs if and only if g(0) > p'(0). F o r / ~ to exist a positive root of cp(~) + q(zT)= g(0) + ~ must exist, i.e. p(fi) = [g(0) + 7]zT/1 + cg must have a positive root. This m a y or may not happen. Finally E* exists if and only if the algebraic system: g(x)=q(u)
(9)
g(x ) + 7 = q(u) + cp(u),
has a positive solution (x*, u*). Then, p ( u * ) = 7 / c and g ( x * ) = q ( u * ) = ? / c u * . Hence, E* exists if and only if 7/c is in the range of p(u) and g(0) > ?/cu*, where
p(u*)=~/c.
We now compute the variational matrices in the standard way (Freedman, 1980): Vo , 1), I7, about the b o u n d a r y equilibria E o ,/~,/~, respectively:
!
;
r~g'(~)
--Lo
p= [cp(a)- ~
o
L aV(o) -a[cp'(a)+q'(a)]1
PREDATOR-PREY SYSTEMS
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F r o m Vo, we note that if g ( 0 ) + 7 - p ' ( 0 ) < 0 , then 9 ( 0 ) - p ' ( 0 ) < 0 and E o is asymptotically stable, implying that persistence cannot occur. In the untransformed system (6) this implies that there are solutions (x(t), y(t))--.(O, + ~ ) as t ~ o o in this case. Hence, we assume:
(A1). g(0)+y--p'(0)>0. E o is then unstable locally in the u direction. We now note that ifp(u)0, then from system (6), p = y ( - 7 + c p ( u ) ) < O for y > 0 , so that lim,_~ y ( t ) = 0 and then either x--*0 or u ~ o o , or both. Hence, we assume the following. (A2). u* > 0 exists such that p(u*) = y/c. Our next assumption is one that is difficult to justify biologically, but seems to be required in order to obtain boundedness and persistence of solutions. (A3). cp'(u)+q'(u)>O. Of course p'(u)>0 always holds. However, for many functional responses (such as Holling types I and II), q'(u)<.O, and so (A3) is non-vacuous. Next we make the assumption that/~ exists, for if not, then we believe that unbounded solutions can occur. (A4). P exists. With the above, we can prove that solutions u(t) are bounded. THEOREM 4. Let (H1)--(H3) and (A1)-(A4) hold. Then M(x0/Y0) exists such that 0 <. u(t) ~ 0( < 0) if E* exists (does not exist). The main result of this section may now be presented. THEOREM 5. Assume that (H1)-(H3) and (A1)-(A4) hold. Then system (8), and hence system (6), persists (does not persist) if E* exists (does not exist). A simple test of the persistence is given in the following corollary. C O R O L L A R Y 1. Assume that (H1)-(H3) and (A1)-(A4) hold. Then system (8), and hence system (6), persists (does not persist) if c p ( f i ) - 7 > 0( < 0). We conclude this section with an example to illustrate our results.
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H . I . F R E E D M A N A N D R. M. MATHSEN
Consider the system:
£= x(c~- ~ x ) - y
( ,+c
x(0) = x o > 0
y(o) = yo > o.
(lO)
T h e n the transformed system becomes:
2=x[e-flx
f i = u ~--flx+~
b+ua ], bacu +u
x(0)=Xo> 0
b +a u 1'
u(0)--x°>0"
Yo
(11)
Hence, 9(x) = ~ - fix, p(u) = au/(b + u), q(u) = a/(b + u). F o r (A1) to hold we will require b(c~+7)>a. For (A2) we require u* > 0 to exist. This in turn requires ac > 7, in which case u* = bv/(ac- 7). F o r (A3) to be valid the inequality bc > 1 must hold. C o m p u t i n g 2 and ~7gives 2 = ~b - a/~b and ~7= [ ( c~ +? )b-a]/ [ac-~-~]. By (A1), 2 exists and ~7 exists if ac>~+~, thereby verifying (A4). Finally cp(a) > ~ is valid (the criterion for persistence) if and only if c~bc+ 7 > ac. As a final note, we observe that all the above inequalities hold when c~= 1,
~=½, a = l , b = l , c = 2 , ~ > O . 4. Discussion. In this paper we have been concerned with p r e d a t o r - p r e y systems with ratio dependence incorporated into the models. It has been our main purpose to obtain criteria for persistence, i.e. criteria which guarantee that solutions with positive initial values do not get too close to the axes, where the models are not valid. Such considerations are important if the analysis of the model is to have any validity. Even if solutions, representing populations close to equilibrium, are considered as initiating well in the interior of the positive quadrant, there is nothing a priori to prevent them from getting close to an axis in a short time, unless persistence is shown. In the predator-to-prey ratio dependence case, under reasonable assumptions on the model, it was shown that all solutions are bounded, that an interior
PREDATOR-PREY SYSTEMS
825
equilibrium exists and that the system is persistent. It is of interest, however, to note in this case that uniform persistence was not and we believe cannot be shown. In the prey-to-predator ratio dependence case, additional hypotheses were required for boundedness of solutions, and the existence of an interior equilibrium was necessary and sufficient for persistence. This last statement coincides with the non-ratio dependence case. The ecological reasons for incorporating ratio dependence into predatorprey models have been argued elsewhere. It was not our purpose to enforce or refute these arguments, but rather to give criteria under which such models can yield to a valid analysis when they are used. We believe that "good" science must contain as a minimum "correct" science, and hence our preoccupation with "proving" persistence in order to establish the "mathematical validity" of ratio-dependent models.
APPENDIX In this appendix we prove our theorems. In order to do so, we state a version of the B u t l e r - M c G e h e e lemma (Freedman and Waltman, 1984) which is suitable for our setting. LEMMA 2. ( B u t l e ~ M c G e h e e ) Let M be an isolated invariant set, f2 an omega limit set of some orbit, and M c ~ . Then if f2v~M, it follows that f 2 c ~ W + \ M ¢ 0 and f~c~W \ M e 0 , where W + ( W - ) is the strong stable (unstable) manifold of M. Proof of Theorem 1. (1) and (2) are obvious. To prove (3) we note that f(z) is a decreasing function and so iff ( 0 ) ~ 0. However, iff ( 0 ) > g(0), sincef(L) = 0, exists such that f ( ~ ) = 9(0). We now prove (4). Clearly (K, 0 ) E F 1 . Further:
dz
dx
p(x)g'(x)-g(x)p'(x) < 0
p(x)2
Fl
on
O
Finally limx~ o + g(x)/p(x) = + oo. As a consequence x* exists, since g(x)/p(x) takes on all values between zero and infinity. Further (x*, L) satisfies F z s i n c e f ( L ) = 0 . To show the uniqueness of E* we note that from equation (4),f(z)= 0 must hold, but this occurs only for z = L. Then, since g(x)/p(x) is strictly decreasing, g(x)/p(x)= L only for x = x*. • Proof of Theorem 2. First note that along solutions of system (3):
dz
z [ f ( z ) - - g ( x ) + zp(x)]
dx
x[g(x)-zp(x)]
Then:
dz
dx
x
K
_ z[f(z) + zp(K)] - Kzp(K)
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H.I. FREEDMAN AND R. M. MATHSEN
Hence: dz
dx
<0 >0
x=K
if if
f(z)+zp(K)>O f(z)+zp(K)
Further dx/dt]x=r=-Kzp(K) m a x ( L , Zk), such that f(Zk) + Zkp(K)= 0. Then such solutions are trapped by themselves in forward time, since system (3) is a dynamical system. Orbits which start "outside" (go(t), ~(t)) have similar properties. Orbits starting "inside" are trapped by (go(t), g,(t)) and so all orbits are bounded, i.e. O<~x(t)<~m(Xo) and 0 ~
V(x, ~= [ g t x ) - zp(x) + x D ' ( x ) - zp'(x)] - xp(x) ]. z, L -zg'(x)+z2p'(x) f(z)-g(x)+zp(x)+z[f'(z)+p(x)]j Let Vo, V1, V2 and V* be the variational matrices about E o, E 1 , E 2 and E*, respectively. Then: 0
- [
,
,,
0(0)
2 - L z [ - 9 ' ( o ) + zp'(O)]
v,=Vx*D'(x*)-Lp'(x*)] L L[Lp'(x*)-g'(x*)]
~f,o(~)],
-x*p(x*) ] L[f'(L)+p(x*)]J"
F r o m these variational matrices we are able to obtain information on their eigenvalues and the stability of the equilibria. Since we are interested in the persistence of system (3), we focus only on the boundary equilibria. Since Vo is triangular, its eigenvalues are g(0) a n d f ( 0 ) - g ( 0 ) . Since g(0)>0, E o is unstable locally along the x-axis. Similarly, V1 has eigenvalues Kg'(K) and f(0). Since Kg'(K) 0, E 1 is stable locally in the x-direction and unstable locally in the z-direction. Finally, when E 2 exists, V2's eigenvalues are g(0) > 0 and zf'(~) < 0, and so E 2 is unstable locally along the x-direction and stable locally along the z-axis. Also note that E o is unstable locally along the z-axis if E 2 exists and stable if it does not. Proof of Theorem 3. The theorem will be proved if we can show that there are no omega limit points on the axes of orbits initiating in the interior of the positive quadrant. Let fl be such an omega limit set. If E 2 exists, then E o is completely unstable and so E o q~~. If E 2 does not exist, then W + (Eo) is the z-axis and so by the Butler-McGehee lemma Q ~ W + (Eo) exists such that Q e ft. But then the closure of the orbit through Q belongs to ~ and this orbit is the positive z-axis, which is unbounded. However, by Theorems 1 and 3, all orbits are bounded in positive time and so fl is bounded. This contradiction shows that E o ¢ f l in all cases. Now suppose E 1 ~fl. E 1 is a saddle point and so Q1 e W+(EO exists as before. However, W+(E1)={(x, z):z=O, (O K , then fl is unbounded. If Q
PREDATOR PREY SYSTEMS
827
Proof of Theorem 4. From (8): fi(t) = u(t ) [g(x(t)) + 7 - cp(u(t))-- q(u(t))] < u(t) [g(0) + y-- cp(u(t))-- q(u(t ))]. Let: F(u) = g(O) + ~ - cp(u)- q(u).
(lo)
F(O)=g(O)+7-p'(O)>O by (A1). F(g)=0 by definition. F ' ( u ) = - c p ' ( u ) - q ' ( u ) < O by (A3). Hence, if u > ~7, F(u)<0 and so fi(t)< 0 proving the theorem. • Proof ofLemma 5. Assume that E* exists. Then 9(O)>q(u*). Consider the function F(u) defined in (10). F(O)=g(O)+7-p'(O)>O by (A1). F(ti)=0 by definition. F ' ( u ) = - c p ' ( u ) q'(u)<0 by (A3). F(u*)=g(O)+7-ep(u*)-q(u*)=g(O)-q(u*) by definition of u*. However, g(O)--q(u*) > 0. Hence, since F(u) is a decreasing function, u*< tT. Therefore, cp(g)-7 > O. The proof for E* not existing is analogous. • The proof of persistence in Theorem 6 when E* exists is similar to the above proof of Theorem 3, and so the details will not be given. The non-persistence when E* does not exist follows readily, since then P is asymptotically stable.
LITERATURE Arditi, R. and L. R. Ginzburg. 1989. Coupling in predator prey dynamics: ratio dependence. J. theor. Biol. 139, 311-326. Butler, G.', H. I. Freedman and P. Waltman. 1986. Uniformly persistent systems. Proc. Am. Math. Soc. 96, 425429. Butler, G. and P. Waltman. 1986. Persistence in dynamical systems. J. Differential Equations 63, 255-263. Freedman, H. I. 1980. Deterministic Mathematical Models in Population Ecology. New York: Marcel Dekker. Freedman, H. I. and P. Moson. 1990. Persistence definitions and their connections. Proc. Am. Math. Soc. 109, 1025-1033. Freedman, H. I. and P. Waltman. 1984. Persistence in models of three interacting predator-prey populations. Math. Biosci. 68, 213-231. Freedman, H. I. and P. Waltman. 1985. Persistence in a model of three competitive populations. Math. Biosci. 73, 89-101. Gard, T. C. 1987. Uniform persistence in multispecies population models. Math. Biosci. 85, 93-104. Gause, G. F., N. P. Smaragdova and A. A. Witt. 1936. Further studies of interaction between predators and prey. J. Anita. Ecol. 5, 1-18. Lindstrom, T. Preprint. Qualitative analysis of a predator-prey system with limit cycles. Sarkar, A. K., D. Mitra, S. Ray and A. B. Roy. 1991. Permanence and oscillatory coexistence of a detrius-based prey-predator model. Ecol. Model 53, 147-156. Veilleux, B. G. 1979. An analysis between the predatory interaction between paramecium and didinium. J. Anim. Ecol. 48, 787-803. Wiens, J. A., J. F. Addicott, T. J. Case and J. Diamond. 1986. Overview: the importance of spatial and temporal scale in ecological investigations. In: Community Ecology. J. Diamond and T. J. Case (Eds), pp. 145-153. New York: Harper and Row.
R e c e i v e d 21 J u l y 1992
0092-8240/9356 00+ 0.00 PergamonPressLtd © 1993 Societyfor Mathematical Biology
PERSISTENCE IN PREDATOR-PREY SYSTEMS WITH RATIO-DEPENDENT PREDATOR INFLUENCE H. I. F R E E D M A N ? Department of Mathematics, Applied Mathematics Institute, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 (E.mail :[email protected] )
R. M. M A T H S E N ~ Department of Mathematics, North Dakota State University, Fargo, ND 58105-5057, U.S.A. Predator-prey models where one or more terms involve ratios of the predator and prey populations may not be valid mathematically unless it can be shown that solutions with positive initial conditions never get arbitrarily close to the axis in question, i.e. that persistence holds. By means of a transformation of variables, criteria for persistence are derived for two classes of such models, thereby leading to their validity. Although local extinction certainly is a common occurrence in nature, it cannot be modeled by systems which are ratio-dependent near the axes.
1. Introduction. Continuous models, usually in the form of differential equations, have formed a large part of the traditional mathematical ecology literature. In order to carry out the required mathematical analyses, they are assumed to be sufficiently smooth over their entire domain of definition so that solutions to initial value problems exist uniquely and are continuable for all positive time. In particular, in the case of predator-prey systems the domain must include the prey and predator axes, since one must know the dynamics of each population in the absence of the other (see, e.g. Freedman, 1980; Freedman and Waltman, 1984 and refs therein). Siinilar statements can be made with respect to competition models, mutualism models or models which incorporate combinations of these (Freedman, 1980; Freedman and Waltman, 1985). Recently, however, several manuscripts have been written, and more are likely to appear, which incorporate terms involving the ratio of prey-topredator or predator-to-prey. These terms have appeared in place of predator functional response terms or in the predator growth terms (Arditi and t Research partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. NSERC A4823. :~ Research carried out while visiting the Uni~cersity of Alberta. 817
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H . I . F R E E D M A N A N D R. M. M A T H S E N
Ginzburg, 1989; Lindstrom, preprint; Sarkar et al., 1991). The ecological reasons for choosing ratio dependence in Arditi and Ginzburg (1989) are related to scale. Wiens et al. (1986) have shown that choosing various temporal or spatial scales can lead to different outcomes. Introducing ratio dependence leads to a reduction of shared resources for the predator if more predators are present in the prey-to-predator ratio case, for example. It is not our purpose in this paper to discuss the ecological validity of such ratio-dependent models, but to note that such ratio dependence does lead to some mathematical problems and these have not been dealt with in the above mentioned manuscripts, and therefore we deal with them here. By virtue of ratio dependence the model is undefined, and hence discontinuous, on either the prey or predator axis. Hence, any analysis of solutions initiating with positive populations is valid only if one can show that such solutions "stay away" from the axes. By "stay away" we mean that there does not exist a sequence of times tending to infinity such that either the prey values or predator values tends to zero. A convenient way of expressing this is in terms of "persistence". In the real world, in a predator-prey setting, persistence of populations in a given environment or local extinction of one or more population may occur. Gause et al. (1936) showed that local extinction could indeed occur after only a few generations in paramecuim-didinium interactions in the laboratory. (However, Veilleux, 1979, in repeating these laboratory interactions was able to sustain both populations for over 50 cycles, until he terminated the experiment.) We feel, however, since the ratio-dependent models are not mathematically correct on the axes, they ought not to be utilized when extinction occurs unless the functional forms of the models are appropriately redefined near the axes. Hence, we are interested in the question "when does persistence occur in ratio-dependent models?" In such cases, the models are indeed valid throughout the positive orthant. In an ecological context, persistence may be defined mathematically as in Freedman and Waltman (1984, 1985). A population N(t) is said to persist (sometimes termed strongly persist) if N(0)> 0 implies N(t)> 0 for t > 0 and lim inft_.o~ N(t)>0. For related definitions in both ecological and abstract situations, see Butler et al. (1986), Butler and Waltman (1986) and Freedman and Moson (1990). One of the related definitions that will be of interest is uniform persistence (also known as permanence). N(t) is said to persist uniformly if N(t) persists and there exists 6 > 0 , independent of N(0)>0, such that lim inft_. 00N(t) >~6. Finally, we say that a system (uniformly) persists whenever each component (uniformly) persists. The standard techniques for showing persistence, which require the system to be defined on the axes, do not work here unless the system can be transformed into a dynamical system. This we are able to do for each of the
PREDATOR-PREY SYSTEMS
819
models of interest. By a change of variables we transform each of the given systems into a dynamical system in such a way that there is a one-to-one correspondence between the positive values of the predator and prey of the original system and the positive values of the transformed system. Hence, persistence in the transformed system implies persistence of predator and prey, i.e. it implies that the analyses in the ratio-dependent manuscripts are valid. The usual techniques for deriving persistence typically also imply uniform persistence. It is of interest to note that such is not the case here. In the cases considered in this paper, the usual techniques described in Freedman and Waltman (1984) or in Gard (1987) do not work, since we cannot prove that the transformed systems are dissipative. However, we are able to obtain persistence criteria utilizing geometrical arguments. In Section 2 we will consider the case where the ratio dependence is predatorto-prey and Section 3 where it is prey-to-predator. In the first case, the standard assumptions will give persistence. However, in the second case, additional assumptions will be required. Both cases are illustrated by examples. We conclude with a short discussion in Section 4. All proofs of our results are deferred to the appendix. 2. Predator-to-Prey Ratio Dependence. predator-prey model of the form:
In this section we consider a
xg(x)-- yp(x), x(0) = x o > 0 y =yf(y/x),
y(0) = Yo > 0;
(1)
where x(t) is the prey density, y(t) is the predator density and "= d/dt. We assume the following: (H1) g, P and f have second derivatives continuous in their arguments on the interval (0, oe). This is sufficient to guarantee that solutions to positive initial value problems exist uniquely at least for some positive ti.me. (H2') 9(0)>0; g'(x)<0; there exists K > 0 such that g(K)=0. For small values of the prey population, it will grow. However, there exists a carrying capacity of the environment beyond which the prey population cannot increase, even in the absence of predators. (H3) p(0)=0; p'(x)>0. The predator functional response is an increasing function of the prey, and, of course, in the absence of prey there can be no predation. Before stating our last hypothesis, we effect the change of variables
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H . I . F R E E D M A N A N D R. M. M A T H S E N
(2)
z =y/x. Then f ( y / x ) = f ( z ) .
(H4) f(0) > 0; f'(z) < 0; there exists L > 0 such that f ( L ) = O. z is small if y is small or x is large, i.e. few predators or many prey. In either case we would expect the predator population to grow. However, if the predator-to-prey ratio increases, it is reasonable to assume that the predator growth rate would decrease to the point where growth changes to loss (at y/x=L). Under these assumptions, our model includes the model of Lindstrom (preprint) and the submodel in Sarkar et al. (1991) representing microorganisms feeding on detritus. We now derive the system satisfied by x and z. By (1) and (2), xzf(z) = ~9= d/dt(xz) = x~ + z2 = x~ + z i g ( x ) - zp(x)]. Hence, system (1) transforms to: x [ g ( x ) - zp(x)], i = z [ f ( z ) - g(x) + zp(x)],
x(0) = x 0 > 0 z ( 0 ) = Yo > 0.
(3)
x0
Note that positive solutions of system (3) are in a one-to-one correspondence with positive solutions of system (1). However, system (3) can be extended to the axes in a smooth manner, thereby causing it to be a dynamical system over R2+, the non-negative quadrant. Further, since persistence deals only with positive solutions, persistence in system (3) implies persistence in system (1). We now note that the prey population in the transformed model continues to be limited by its resources, i.e. x(t) is bounded. In fact, lim supt_.o~ x(t)<~ K. Next we examine the existence of equilibria in system (3), i.e. those values of x and z for which 2 -- ~ = 0. Algebraically, this involves setting the right sides of system (3) equal to zero and solving. Geometrically, positive equilibria occur whenever the isoclines intersect, where the isoclines are the curves given by: r l :z = g(x)/p(x)
F 2 :f(z) + zp(x) = O(x).
(4)
We now have the following results on the existence of equilibria. THEOREM 1. System (3) has at most four equilibria as follows: (1) E0(0, 0) always exists. (2) E 1(K, 0) always exists. (3) E2(0, ~), where f(~)=g(0) and 0 < ~ < L , exists if and only/ff(0)>g(0).
PREDATOR-PREY SYSTEMS
(4) E*(x*, L), where g(x*)=Lp(x*) and 0 < x * < K , unique.
821
always exists and is
In order to obtain our next result, which is p a r a m o u n t in proving persistence for system (3), we require some information about F 2. I f f ( 0 ) > g ( 0 ) , then (0, ~) e F 2 . If f(0) = g(0), then (0, 0) e F 2 . If f(0) < g(0), there exists a point of the form ()~, 0 ) ~ F 2, where g ( Y ) = f ( 0 ) > 0 and hence 0 < X < K . N o w consider the case where z = L. Since using equation (4) the only value of x that satisfies - - g ( x ) + L p ( x ) = O is x = x * , F 2 can cross z = L only at E*. Hence, the curve F2, once it rises above L, cannot come below it again. THEOREM 2. Let f(z) + zp(K) = 0 have at most a finite number of roots. Then z(t) is bounded for t >~O. We can now state our main result of this section. THEOREM 3. Let hypotheses (H1)-(H4) hold. Further, assume that f(z)+ zp(K) = 0 has at most a finite number of positive roots. Then system (3), and hence system (1), exhibits persistence. We now illustrate the above results by means of an example. We consider the system: 2=x(a-bx
))=y(d-~),
c~x)'
x(0)=x°>0
y(0)=yo>0.
(5)
Here, g(x) = x ( a - bx), p(x) = cx/a + x, f ( z ) = d - ez, K = a/b and L = die. We note that hypotheses (H1)-(H4) are all satisfied. We now observe that the functionf(z) + zp(K) = (ac - ae-- abe/a + ab )z + d has at most one positive zero. Hence, by T h e o r e m 4 all solutions of (5) persist and therefore any analysis of positive solutions is valid.
3. Prey-to-Predator Ratio Dependence. I n
this section we consider a
p r e d a t o r - p r e y system of the form: x(0) = Xo > 0
y(0) = Yo > 0; where 7, c > 0.
(6)
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H . I . F R E E D M A N A N D R. M. M A T H S E N
Here, we let u = x / y and we assume (H1)-(H3) hold, replacing z by u in p. This is precisely the model considered by Arditi and Ginzburg (1989). We n o w derive the transformed system for the variables x and u: fi = (y2 - x~ )/y 2 = (xg(x) - yp(u) + 7x - cxp(u) )/y = u g ( u ) - p(u) + 7 u - cup(u).
If we set:
q(u)=
p(u) u'
u>0
(7)
p'(0), u=0; then our model becomes: 2 = x[g(x) -- q(u)],
x(0) = x o > 0
fi = u[g(x) + ~ -- cp(u)-- q(u)],
u(0) = x° > 0. Yo
(8)
We note that for system (6) O < < . x ( t ) ~ m ( x o ) = m a x { x o, K}. F o r system (6) there are at most four non-negative equilibria: Eo(0, 0), /~(2, 0), /2(0, ~7) and E * ( x * , u*). F o r /~ to exist 2 > 0 must exist such that g(2) = p'(0). This occurs if and only if g(0) > p'(0). F o r / ~ to exist a positive root of cp(~) + q(zT)= g(0) + ~ must exist, i.e. p(fi) = [g(0) + 7]zT/1 + cg must have a positive root. This m a y or may not happen. Finally E* exists if and only if the algebraic system: g(x)=q(u)
(9)
g(x ) + 7 = q(u) + cp(u),
has a positive solution (x*, u*). Then, p ( u * ) = 7 / c and g ( x * ) = q ( u * ) = ? / c u * . Hence, E* exists if and only if 7/c is in the range of p(u) and g(0) > ?/cu*, where
p(u*)=~/c.
We now compute the variational matrices in the standard way (Freedman, 1980): Vo , 1), I7, about the b o u n d a r y equilibria E o ,/~,/~, respectively:
!
;
r~g'(~)
--Lo
p= [cp(a)- ~
o
L aV(o) -a[cp'(a)+q'(a)]1
PREDATOR-PREY SYSTEMS
823
F r o m Vo, we note that if g ( 0 ) + 7 - p ' ( 0 ) < 0 , then 9 ( 0 ) - p ' ( 0 ) < 0 and E o is asymptotically stable, implying that persistence cannot occur. In the untransformed system (6) this implies that there are solutions (x(t), y(t))--.(O, + ~ ) as t ~ o o in this case. Hence, we assume:
(A1). g(0)+y--p'(0)>0. E o is then unstable locally in the u direction. We now note that ifp(u)
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H . I . F R E E D M A N A N D R. M. MATHSEN
Consider the system:
£= x(c~- ~ x ) - y
( ,+c
x(0) = x o > 0
y(o) = yo > o.
(lO)
T h e n the transformed system becomes:
2=x[e-flx
f i = u ~--flx+~
b+ua ], bacu +u
x(0)=Xo> 0
b +a u 1'
u(0)--x°>0"
Yo
(11)
Hence, 9(x) = ~ - fix, p(u) = au/(b + u), q(u) = a/(b + u). F o r (A1) to hold we will require b(c~+7)>a. For (A2) we require u* > 0 to exist. This in turn requires ac > 7, in which case u* = bv/(ac- 7). F o r (A3) to be valid the inequality bc > 1 must hold. C o m p u t i n g 2 and ~7gives 2 = ~b - a/~b and ~7= [ ( c~ +? )b-a]/ [ac-~-~]. By (A1), 2 exists and ~7 exists if ac>~+~, thereby verifying (A4). Finally cp(a) > ~ is valid (the criterion for persistence) if and only if c~bc+ 7 > ac. As a final note, we observe that all the above inequalities hold when c~= 1,
~=½, a = l , b = l , c = 2 , ~ > O . 4. Discussion. In this paper we have been concerned with p r e d a t o r - p r e y systems with ratio dependence incorporated into the models. It has been our main purpose to obtain criteria for persistence, i.e. criteria which guarantee that solutions with positive initial values do not get too close to the axes, where the models are not valid. Such considerations are important if the analysis of the model is to have any validity. Even if solutions, representing populations close to equilibrium, are considered as initiating well in the interior of the positive quadrant, there is nothing a priori to prevent them from getting close to an axis in a short time, unless persistence is shown. In the predator-to-prey ratio dependence case, under reasonable assumptions on the model, it was shown that all solutions are bounded, that an interior
PREDATOR-PREY SYSTEMS
825
equilibrium exists and that the system is persistent. It is of interest, however, to note in this case that uniform persistence was not and we believe cannot be shown. In the prey-to-predator ratio dependence case, additional hypotheses were required for boundedness of solutions, and the existence of an interior equilibrium was necessary and sufficient for persistence. This last statement coincides with the non-ratio dependence case. The ecological reasons for incorporating ratio dependence into predatorprey models have been argued elsewhere. It was not our purpose to enforce or refute these arguments, but rather to give criteria under which such models can yield to a valid analysis when they are used. We believe that "good" science must contain as a minimum "correct" science, and hence our preoccupation with "proving" persistence in order to establish the "mathematical validity" of ratio-dependent models.
APPENDIX In this appendix we prove our theorems. In order to do so, we state a version of the B u t l e r - M c G e h e e lemma (Freedman and Waltman, 1984) which is suitable for our setting. LEMMA 2. ( B u t l e ~ M c G e h e e ) Let M be an isolated invariant set, f2 an omega limit set of some orbit, and M c ~ . Then if f2v~M, it follows that f 2 c ~ W + \ M ¢ 0 and f~c~W \ M e 0 , where W + ( W - ) is the strong stable (unstable) manifold of M. Proof of Theorem 1. (1) and (2) are obvious. To prove (3) we note that f(z) is a decreasing function and so iff ( 0 ) ~
dz
dx
p(x)g'(x)-g(x)p'(x) < 0
p(x)2
Fl
on
O
Finally limx~ o + g(x)/p(x) = + oo. As a consequence x* exists, since g(x)/p(x) takes on all values between zero and infinity. Further (x*, L) satisfies F z s i n c e f ( L ) = 0 . To show the uniqueness of E* we note that from equation (4),f(z)= 0 must hold, but this occurs only for z = L. Then, since g(x)/p(x) is strictly decreasing, g(x)/p(x)= L only for x = x*. • Proof of Theorem 2. First note that along solutions of system (3):
dz
z [ f ( z ) - - g ( x ) + zp(x)]
dx
x[g(x)-zp(x)]
Then:
dz
dx
x
K
_ z[f(z) + zp(K)] - Kzp(K)
826
H.I. FREEDMAN AND R. M. MATHSEN
Hence: dz
dx
<0 >0
x=K
if if
f(z)+zp(K)>O f(z)+zp(K)
Further dx/dt]x=r=-Kzp(K)
V(x, ~= [ g t x ) - zp(x) + x D ' ( x ) - zp'(x)] - xp(x) ]. z, L -zg'(x)+z2p'(x) f(z)-g(x)+zp(x)+z[f'(z)+p(x)]j Let Vo, V1, V2 and V* be the variational matrices about E o, E 1 , E 2 and E*, respectively. Then: 0
- [
,
,,
0(0)
2 - L z [ - 9 ' ( o ) + zp'(O)]
v,=Vx*D'(x*)-Lp'(x*)] L L[Lp'(x*)-g'(x*)]
~f,o(~)],
-x*p(x*) ] L[f'(L)+p(x*)]J"
F r o m these variational matrices we are able to obtain information on their eigenvalues and the stability of the equilibria. Since we are interested in the persistence of system (3), we focus only on the boundary equilibria. Since Vo is triangular, its eigenvalues are g(0) a n d f ( 0 ) - g ( 0 ) . Since g(0)>0, E o is unstable locally along the x-axis. Similarly, V1 has eigenvalues Kg'(K) and f(0). Since Kg'(K)
PREDATOR PREY SYSTEMS
827
Proof of Theorem 4. From (8): fi(t) = u(t ) [g(x(t)) + 7 - cp(u(t))-- q(u(t))] < u(t) [g(0) + y-- cp(u(t))-- q(u(t ))]. Let: F(u) = g(O) + ~ - cp(u)- q(u).
(lo)
F(O)=g(O)+7-p'(O)>O by (A1). F(g)=0 by definition. F ' ( u ) = - c p ' ( u ) - q ' ( u ) < O by (A3). Hence, if u > ~7, F(u)<0 and so fi(t)< 0 proving the theorem. • Proof ofLemma 5. Assume that E* exists. Then 9(O)>q(u*). Consider the function F(u) defined in (10). F(O)=g(O)+7-p'(O)>O by (A1). F(ti)=0 by definition. F ' ( u ) = - c p ' ( u ) q'(u)<0 by (A3). F(u*)=g(O)+7-ep(u*)-q(u*)=g(O)-q(u*) by definition of u*. However, g(O)--q(u*) > 0. Hence, since F(u) is a decreasing function, u*< tT. Therefore, cp(g)-7 > O. The proof for E* not existing is analogous. • The proof of persistence in Theorem 6 when E* exists is similar to the above proof of Theorem 3, and so the details will not be given. The non-persistence when E* does not exist follows readily, since then P is asymptotically stable.
LITERATURE Arditi, R. and L. R. Ginzburg. 1989. Coupling in predator prey dynamics: ratio dependence. J. theor. Biol. 139, 311-326. Butler, G.', H. I. Freedman and P. Waltman. 1986. Uniformly persistent systems. Proc. Am. Math. Soc. 96, 425429. Butler, G. and P. Waltman. 1986. Persistence in dynamical systems. J. Differential Equations 63, 255-263. Freedman, H. I. 1980. Deterministic Mathematical Models in Population Ecology. New York: Marcel Dekker. Freedman, H. I. and P. Moson. 1990. Persistence definitions and their connections. Proc. Am. Math. Soc. 109, 1025-1033. Freedman, H. I. and P. Waltman. 1984. Persistence in models of three interacting predator-prey populations. Math. Biosci. 68, 213-231. Freedman, H. I. and P. Waltman. 1985. Persistence in a model of three competitive populations. Math. Biosci. 73, 89-101. Gard, T. C. 1987. Uniform persistence in multispecies population models. Math. Biosci. 85, 93-104. Gause, G. F., N. P. Smaragdova and A. A. Witt. 1936. Further studies of interaction between predators and prey. J. Anita. Ecol. 5, 1-18. Lindstrom, T. Preprint. Qualitative analysis of a predator-prey system with limit cycles. Sarkar, A. K., D. Mitra, S. Ray and A. B. Roy. 1991. Permanence and oscillatory coexistence of a detrius-based prey-predator model. Ecol. Model 53, 147-156. Veilleux, B. G. 1979. An analysis between the predatory interaction between paramecium and didinium. J. Anim. Ecol. 48, 787-803. Wiens, J. A., J. F. Addicott, T. J. Case and J. Diamond. 1986. Overview: the importance of spatial and temporal scale in ecological investigations. In: Community Ecology. J. Diamond and T. J. Case (Eds), pp. 145-153. New York: Harper and Row.
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