Theoretical Population Biology 55, 111126 (1999) Article ID tpbi.1998.1399, available online at http:ÂÂwww.idealibrary.com on
Optimal Foraging and PredatorPrey Dynamics, II
Vlastimil Kr ivan and Asim Sikder Department of Theoretical Biology, Institute of Entomology, Academy of Sciences of the Czech Republic, and Faculty of Biological Sciences USB, Branis ovska 31, 370 05 C8 eske Bude jovice, Czech Republic E-mail: krivanÄentu.cas.cz
Received May 11, 1997
In this paper we consider one-predatortwo-prey population dynamics described by a control system. We study and compare conditions for permanence of the system for three types of predator feeding behaviors: (i) specialized feeding on the more profitable prey type, (ii) generalized feeding on both prey types, and (iii) optimal foraging behavior. We show that the region of parameter space leading to permanence for optimal foraging behavior is smaller than that for specialized behavior, but larger than that for generalized behavior. This suggests that optimal foraging behavior of predators may promote coexistence in predatorprey systems. We also study the effects of the above three feeding behaviors on apparent competition between the two prey types. ] 1999 Academic Press Key Wordsy optimal foraging; population dynamics; apparent competition; persistence; permanence.
INTRODUCTION profitable prey type is more gradual than predicted by optimal foraging theory. In Kr ivan (1996) a two-preyone-predator population Kr ivan (1996) showed that optimal foraging leads dynamic model with optimal predator foraging behavior naturally to a more general class of population dynami- was considered. This model assumes that predators cal systems which are described by differential inclusions forage according to optimal foraging theory (Charnov, (i.e., by differential equations with multivalued right- 1976; Stephens and Krebs, 1986), which predicts that the hand sides), and to the emergence of partial preferences more profitable prey type is always included in the for the alternative prey type as a consequence of popula- predator diet while the less profitable (i.e., alternative) tion dynamics. The multivaluedness in the model des- prey type is included with probability one only if the den- cription is due to the non-uniqueness of optimal foraging sity of the more profitable prey type falls below a critical strategy when the more profitable prey type reaches the threshold. Therefore, the optimal foraging model does critical threshold. It was shown that the interplay among not predict the emergence of partial preferences for the behavioral ecology and population dynamics is twofold: alternative prey type which is either completely included optimal foraging influences population dynamics, which, or excluded from predators' diet. The predictions of in turn, affects optimal prey diet. Analysis given in optimal foraging theory were compared with several Kr ivan (1996) focused mainly on local stability of the experimental and field studies in Stephens and Krebs ecological equilibrium and analytical results given there (1986) (see also Richardson and Verbeeck (1992)). These were only for a special case in which both prey types had studies support the idea that the diet choice is based on the same intrinsic per capita growth rate. The reason for food profitability although the inclusion of the less this limitation was due to the fact that even local stability
111 0040-5809Â99 K30.00 Copyright ] 1999 by Academic Press All rights of reproduction in any form reserved. 112 Kr @van and Sikder analysis for the general case gives very complex expres- predators follow optimal diet choice and predatorprey sions which cannot be efficiently analyzed. On the population dynamics is of the RosenzweigMacArthur numerical evidence it was conceived there that optimal type. foraging may lead to permanence in the one-predator two-prey model; i.e., all three populations can coexist indefinitely. Optimal foraging can also reduce amplitude POPULATION DYNAMICS of fluctuations in population densities when compared with generalist predator behavior. In this part we survey the model introduced in Kr ivan In this paper we study analytically the effect of optimal (1996) where more details can be found. We consider foraging on permanence of the one-predatortwo-prey predators foraging on two prey types. Predator density is system introduced in Kr ivan (1996). We show that the denoted by x , and prey densities are x and x , respec- conditions for permanence are a great deal simpler and 3 1 2 tively. Population dynamics is described by a system of more efficiently tractable than those for local asymptotic differential equations stability. To conclude that all populations can coexist, permanence theory (Hofbauer and Sigmund, 1984; x1 u1 *1 x1 x3 Butler and Waltman, 1986; Hutson and Schmitt, 1992) x$1 =a1 x1 1& & does not require consideration of the complicated \ K1 + 1+u1 h1 *1 x1+u2 h2 *2 x2 behavior of interior (i.e., all populations present) orbits x2 u2 *2 x2 x3 of the model found in Kr ivan (1996). Permanence means x$2=a2 x2 1& & (1) K 1+u h * x +u h * x that there is a lower positive bound such that in a long \ 2 + 1 1 1 1 2 2 2 2 term run all population densities (initially positive) will u1 e1 *1 x1 x3+u2 e2 *2 x2 x3 be above this bound and no population density tends to x$3= &mx3 . 1+u1 h1 *1 x1+u2 h2 *2 x2 infinity. In Butler and Waltman (1986) the results con- cerning permanence are given for continuous dynamical Here u denotes the probability that a predator will systems, i.e., for systems in which trajectories depend i attack the prey type i, * is the search rate of a predator continuously on initial data. Since the system studied in i for the ith prey type, e is the expected net energy gained Kr ivan (1996) is described by differential inclusions i from the ith prey type, and h is the expected handling which define a continuous dynamical system, we can use i time spent with the ith prey type. In what follows we the results of Butler and Waltman (1986) to study its per- assume that either u and u are fixed if predators show manence. 1 2 fixed preferences for their prey or they are chosen accord- We consider the effects of three types of predator ing to the optimal diet model. Maximization of the net foraging behaviors: (i) predators specialize on the more energy intake rate which is used as the fitness measure in profitable prey type only, (ii) predators are generalists optimal foraging theory (Stephens and Krebs, 1986; and they feed on both prey types, and (iii) predators Kr ivan, 1996) is equivalent to maximization of the forage adaptively following rules of optimal foraging instantaneous per capita predator growth rate x$ Âx .We theory. We derive and compare permanence conditions 3 3 remark that model (1) does not consider any direct com- for the system incorporating the above three predator petition between the two prey populations, but the two foraging behavior types. We show that the region of the prey types are in apparent competition through the parameter space for which permanence holds for adap- shared predation (Holt, 1977). Of course, this apparent tive foragers is larger than that for non-adaptive competition appears only provided that both prey types generalist foragers, but smaller than that for specialist are included in predators' diet. foragers which feed only on the more profitable prey Throughout the paper we will assume that the first type. prey type is more profitable for predators than the We do not consider any direct competition among the second alternative prey type, by which we mean that two prey types. Therefore, in the predator absence both prey populations coexist at their equilibrium levels. e e When predators are introduced, predator-mediated 1 > 2 . apparent competition (Holt, 1977) among prey popula- h1 h2 tions may drive on prey type to extinction. For generalist predators the indirect effects of predation on competition If predators follow optimal diet choice, the optimal among the two prey types were studied by Holt (1977). strategy of a predator when encountering a prey depends In this paper we extend this study to the case where on the density of the more profitable prey type which is Optimal Foraging 113 always attacked upon an encounter, i.e., u =1. The x * x x 1 x$ =a x 1& 1 & 1 1 3 alternative prey type is attacked with probability one 1 1 1 \ K1 + 1+h1 *1 x1 (u1=1) if the density of the more profitable prey type is x2 below the critical threshold x$2=a2 x2 1& (3) \ K2 +
e2 e1 *1 x1 x3 x*1= . x$3= &mx3 . *1(e1 h2&e2 h1) 1+h1 *1 x1
Note that the equation for x in (3) is independent of the If the first prey type density is above x* then the alter- 2 1 other two equations. native prey type is not attacked upon an encounter In G0 the right-hand side of (1) is not uniquely defined (u =0) since it pays off for predators to search for the 2 since 0 u 1. However, it was shown in Kr ivan (1996) more profitable prey type. Following Murdoch and 2 that in the subregion of G0 described by Oaten (1975) we call x*1 the switching density because predators switch their behavior at this prey density. If the a x* first prey type density equals x* then optimal predator 1 1& 1 (1+h * x*) 1 * K 1 1 1 strategy is not uniquely defined by maximization of 1 \ 1 + x$ Âx , i.e, 0 u 1. Thus, control u as a function of the a x* 3 3 2 2 1 3 e1 G =[x # R | x1 x*1 x$2=x2 a2 1& x1 *1 x1 x3 \ K1 + x$1 =a1 x1 1& & \ K1 + 1+h1 *1 x1+h2 *2 x2 1 x*1 & *1 x3&a1(1+h1 *1 x*1 ) 1& h2 *1 \ \ K1 ++ x2 *2 x2 x3 x$2=a2 x2 1& & (2) \ K2 + 1+h1 *1 x1+h2 *2 x2 e2 x$3=x3 &m . \h2 + e1 *1 x1 x3+e2 *2 x2 x3 x$3= &mx3 . 1+h1 *1 x1+h2 *2 x2 We want to study permanence of (1) for various predator foraging behaviors. Permanence is a weaker In G2 the more profitable prey type density is high, the notion than stability because it says only that there is a alternative prey type is excluded from the predator diet positive lower and upper bound for population densities (u2=0), and the corresponding dynamics is but it does not give any information regarding qualitative 114 Kr @van and Sikder behavior of trajectories. For deriving conditions for the its profitability measured by the ratio ei Âhi . Population permanence of three-dimensional systems it is enough to dynamics are described by system (2) which has the study the behavior of trajectories of (1) with one popula- following boundary equilibria (i.e., equilibria at which at tion missing. This simplifies the analysis, by reducing the least one population is missing): E0=(0, 0, 0), E1= dimensionality of the problem. Butler and Waltman (K1 ,0,0),E2=(0, K2 ,0),E12=(K1 , K2 ,0), (1986) derived the necessary and sufficient conditions for permanence of a dynamical system. The conditions m E = ,0, require that trajectories of the system depend con- 13 * (e &h m) tinuously on initial data, trajectories of the system are \ 1 1 1 uniformly bounded, all boundary invariant sets repel a e (e K * &(1+h K * ) m) 1 1 1 1 1 1 1 1 , interior trajectories, there are no cycles formed by subsets 2 2 K1 *1(e1&h1 m) + of boundary invariant sets, and all boundary invariant sets are isolated. For Gause-type models such as those m E23= 0, , described by (2) the dynamics when one population is \ *2(e2&h2 m) missing are simple. The two-dimensional x &x (i= i 3 a e (e K * &(1+h K * ) m) 1, 2) predatorprey dynamics when one prey type is mis- 2 2 2 2 2 2 2 2 2 2 . sing has one equilibrium which either is globally K2 *2(e2&h2 m) + asymptotically stable or is unstable and then a globally asymptotically stable limit cycle around this equilibrium We will call E12 , E13 ,andE23 planar equilibria. Since exists (Hofbauer and Sigmund, 1984; Kuang and Freedman, 1988). If the equilibrium is not feasible (by feasible we ei Ki *i ei mean that all coordinates are positive), then due to a < , 1+hi Ki *i hi high mortality rate predator population is eliminated from the two-dimensional system. When the predator equilibrium E (i=1, 2) is feasible (i.e., ith and third population is missing then the two-prey system has one i3 coordinates are positive) if equilibrium (K1 , K2) which is globally asymptotically stable. The conditions ensuring permanence for three- e K * dimensional Gause-type models reduce to verify that the m< i i i , i=1, 2. (4) boundary equilibria are repelling interior trajectories if 1+hi Ki *i the boundary limit cycles do not appear, since other con- ditions of the Butler and Waltman theorem are satisfied. Condition (4) is more likely to be satisfied for large We remark that the conditions ensuring repellence of values of carrying capacity Ki . If the predator mortality boundary equilibria are also called invasibility conditions, rate is higher than the prey profitability measured by because they imply that if the one-preyone-predator ei Âhi , then Ei3 is not feasible. subsystem is in equilibrium and the missing prey is intro- System (2) has one interior equilibrium duced in a small quantity then it will invade the com- munity (Holt, 1977, Hofbauer and Sigmund, 1984). K (a * m&K * (a * &a * )(e &h m)) E (2)= 1 2 1 2 2 2 1 1 2 2 2 , Verification of boundary limit cycle repellence, if it a K *2(e &h m)+a K *2(e &h m) exists, is a cumbersome problem as we do not know the \ 2 1 1 1 1 1 2 2 2 2 analytical description of the limit cycle and thus we can- K (a * m+K * (a * &a * )(e &h m)) 2 1 2 1 1 2 1 1 2 1 1 , not compute the Floquet multipliers which determine 2 2 a2 K1 *1(e1&h1 m)+a1 K2 *2(e2&h2 m) limit cycle repellence. Throughout this paper we will assume that if a boundary limit cycle exists then it repels (e1 K1 *1+e2 K2 *2&(1+h1 K1 *1+h2 K2 *2) m) interior trajectories. This condition can be verified 2 (a1 a2(a1 K2 *2(e2+e2 h1 K1 *1&e1 h2 K1 *1) numerically for each set of parameters. 2 +a2 K1 *1(e1&e2 h1 K2 *2+e1 h2 K2 *2))) 2 2 2 . (a2 K1 *1(e1&h1 m)+a1 K2 *2(e2&h2 m)) + PERMANENCE FOR GENERALIST PREDATORS If predators are missing then equilibrium E12 is always feasible and globally asymptotically stable for the two- Here we assume that predators are generalists and they prey system because densities of both prey types con- forage on every prey encountered (u2=1), regardless of verge to their carrying capacities. If one of the two prey Optimal Foraging 115 types is missing and the corresponding Rosenzweig Invasibility condition (8) can be rewritten as MacArthur predatorprey system has a feasible equi- librium E (i=1, 2) then this equilibrium is globally i3 a2 *1 m asymptotically stable provided that *1 &1 >& (10) \a1 *2 + (e1&mh1) K1 e (K * h &1) m i i i i (5) and condition (9) as hi (Ki *i hi+1) a * m (Hofbauer and Sigmund, 1984). If the opposite in- * 1& 1 2 < . (11) 2 a * (e &mh ) K equality holds in (5), then there is a unique globally \ 2 1 + 2 2 2 asymptotically stable limit cycle around Ei3 (Kuang and Freedman, 1988). In what follows we will assume that The prey type with higher ratio ai Â*i is called the either no boundary limit cycle exists (which happens if keystone species (Paine, 1969; Holt, 1977; Vandermeer and Maruca, 1998). It follows that the keystone prey type either E13 and E23 are not feasible or (5) holds for i=1, 2) or if a boundary limit cycle exists then it repels will always invade the equilibrial community consisting interior trajectories. Under this assumption permanence of the other prey type and predators. For example, if the for (2) holds if all feasible boundary equilibria are repel- first prey type is the keystone species (a1 Â*1>a2 Â*2)then ling interior orbits; i.e., when one population is missing it will always invade the system consisting of the second and the corresponding two-dimensional system is in prey type and predators because condition (11) will hold. equilibrium, then the missing population will invade the Under the same assumption condition (10) constrains community when introduced in small quantity; see the possibility for the second prey type to invade the Appendix A. Predators will invade the two-prey com- community consisting of the first prey type and munity which is at equilibrium E12 if predators. If, for example, the predator mortality rate m is low or carrying capacity K1 is high the second prey type cannot invade. This implies that in this case the e1 K1 *1+e2 K2 *2 m< .(6)second prey type is in higher danger of being eliminated 1+h K * +h K * 1 1 1 2 2 2 from the community when environmental productivity increases, or predator mortality rate decreases. If If opposite inequality holds in (6) then predators will be a Â* =a Â* , which is the case considered in Kr ivan always driven to extinction due to a high mortality rate. 1 1 2 2 (1996), then invasibility conditions (10) and (11) are Note that since the first prey type is more profitable than automatically satisfied. Invasibility conditions with the second prey type the following inequality is satisfied handling times set to zero (i.e., population dynamics is described by the LotkaVolterra model) are discussed in e K * e K * +e K * 2 2 2 < 1 1 1 2 2 2 ,(7)detail in Holt (1977). 1+h2 K2 *2 1+h1 K1 *1+h2 K2 *2 It is known that permanence implies automatically feasibility (densities of all populations are positive) of the (2) which implies that feasibility of equilibrium E23 auto- interior equilibrium E (Hofbauer and Sigmund, 1984). matically implies that predators will invade the com- In our case this can be easily verified directly. Indeed, munity of the two prey types. If E13 is feasible then the repellence of the boundary equilibria E13 and E23 implies second prey type will invade the equilibrial community positivity of both prey densities in equilibrium E (2). consisting of the first prey type and predators if Positivity of the predator equilibrial density follows from the repellence conditions and condition (4). Local (2) ma1 *2 stability analysis of E leads to very complex expres- a2 *1&a1 *2>& (8) sions which do not seem to be readily interpretable. (e1&mh1) K1 *1 To get permanence for (2) we have the following cases with respect to the predator mortality rate; see Appen- and, similarly, if E is feasible then the first prey type will 23 dix A: invade the equilibrial community consisting of the second prey type and predators if (a) ma2 *1 e1 K1 *1 e2 K2 *2 a2 *1&a1 *2< .(9) 0 This is the case in which both predatorprey systems In this case neither E13 nor E23 is feasible, E12 repels inte- with one prey type missing do coexist because both rior trajectories, and no additional assumptions are planar equilibria E13 and E23 are feasible (Fig. 1A). Due needed to have permanence of (2) (Fig. 1D). to (7) equilibrium E12 repels interior trajectories, and E13 The above results can be interpreted from the com- and E23 repel interior orbits if munity ecology point of view. The case (a) considers the situation in which predators can coexist with each of the ma1 *2 two prey types alone. Introduction of the missing prey & e2 K2 *2 alone, but coexistence is possible when both prey types 1+h2 K2 *2 are in the community. Thus we see that conditions which ensure permanence of the three-dimensional system e K * e K * +e K * (d) e2 K2 *2 K1< . (13) *1(e1+K2 *2(e1 h2&e2 h1)) e1 K1 *1 e2 K2 *2 max , Because {1+h1 K1 *1 1+h2 K2 *2 = e1 K1 *1+e2 K2 *2 e2 K2 *2 it follows that (13) can be satisfied only if the carrying capacity of the more profitable prey type is lower than the switching density, i.e., K1 e K * e K * e K * +e K * 2 2 2 < 1 1 1 < 1 1 1 2 2 2 . 1+h2 K2 *2 1+h1 K1 *1 1+h1 K1 *1+h2 K2 *2 This is satisfied if e2 K2 *2 The parameter region leading to permanence looks as in Fig. 2B. If the carrying capacity of the more profitable prey type is higher than the switching density then e K * e K * +e K * 1 1 1 > 1 1 1 2 2 2 1+h1 K1 *1 1+h1 K2 *1+h2 K2 *2 and the region of the parameter space where permanence holds is shown in Fig. 2C. PERMANENCE FOR SPECIALIST PREDATORS Here we consider the case in which predators specialize only on the more profitable prey type and the alternative prey type is excluded from predator diet (u2=0). Pop- ulation dynamics is then described by system (3), which has two planar equilibria E12 and E13 . These two equi- libria are the same as those for system (2). Again, we assume that if the boundary limit cycle exists then it repels interior trajectories. If E13 is feasible then it always repels interior orbits, and E12 repels interior trajectories if e * K m< 1 1 1 ; (15) 1+h1 *1 K1 FIG. 1. This figure shows four possible scenarios that lead to per- manence of (2). In order for (2) to be permanent all planar equilibria must repel interior orbits. In (A) the three planar equilibria E12 , E13 , and E23 are feasible and they are repelling interior trajectories. In (B) and (C) only one of the two planar equilibria E13 and E23 is feasible and repelling. In (D) E13 and E23 do not exist. In all cases it is trivial to see that boundary equilibria are acyclic. File: 653J 139907 . By:SD . Date:01:04:99 . Time:12:47 LOP8M. V8.B. Page 01:01 Codes: 3233 Signs: 1699 . Length: 54 pic 0 pts, 227 mm 118 Kr @van and Sikder File: 653J 139908 . By:SD . Date:01:04:99 . Time:12:48 LOP8M. V8.B. Page 01:01 Codes: 630 Signs: 21 . Length: 54 pic 0 pts, 227 mm Optimal Foraging 119 see Appendix B. Since (15) is equivalent with feasibility of eaten, can always invade the community. We will con- E13 , it follows that (3) is permanent if and only if E13 is sider two cases depending on the position of the switch- feasible. Note that feasibility of E13 is equivalent with the ing threshold x*1 with respect to the carrying capacity K1 existence of a feasible interior equilibrium of the more profitable prey type. First we assume that K1 e K * e e K * +e K * 2 2 2 < 2 < 1 1 1 2 2 2 1+h K * h 1+h K * +h K * PERMANENCE FOR OPTIMALLY 2 2 2 2 1 1 1 2 2 2 FORAGING PREDATORS e1 K1 *1 e1 < < . (16) 1+h1 K1 *1 h1 Now we consider the case in which predators follow optimal prey choice. We show that the parameter space In turn, this ordering implies the occurrence of the that leads to permanence of (1) driven by optimal forag- following cases which lead to permanence; see ing strategy is larger than for the case in which predators Appendix C: are generalists and they choose their diet in a non-adap- (Aa) tive way. As in the previous parts we assume that if a boundary e K * limit cycle exists then it repels interior trajectories. The 0 FIG. 2. In this figure the regions of parameter space that lead to permanence for generalist predators are shown for various values of carrying capacities. Here m1=e2 K2 *2Â(1+h2 K2 *2), m2=e2Âh2 , m3=(e1 K1 *1+e2 K2 *2)Â(1+h1 K1 *1+h2 K2 *2), and m4=e1K1 *1 Â(1+h1 K1 *1), where for this plot the following values of parameters were chose: a1=1, a2=2, e1=4, e2=1, h1=h2=0.5, and *1=*2=2. This choice of parameters satisfies condition (5); i.e., no predatorprey cycles appear when one prey type is missing. In (A) carrying capacities satisfy (13) (K1=0.05, K2=1). In (B) carrying capacities satisfy (14) (K1=0.2, K2=1), and in (C) the more profitable prey type carrying capacity is above the switching threshold; i.e., K1>x*1 (K1=1, K2=1). 120 Kr @van and Sikder FIG. 3. In this figure the region of parameter space that leads to permanence for specialist predators is shown for the same parameters that are used in Fig. 2C. provided that E13 and E23 repel interior orbits, which for permanence for generalist predators. These condi- happens if (12) holds (Fig. 4A). tions ensure that neither of the two prey types will be (Ab) driven to extinction by predation. When predator mor- tality rate is above m2 , for optimally foraging predators planar equilibrium E23 is not feasible and E13 always e K * e 2 2 2 File: 653J 139910 . By:SD . Date:01:04:99 . Time:12:48 LOP8M. V8.B. Page 01:01 Codes: 3986 Signs: 2863 . Length: 54 pic 0 pts, 227 mm Optimal Foraging 121 DISCUSSION In this paper we studied the effects of three types of predator behavior on the permanence of one-predator two-prey population dynamics. These types of behavior were: (i) predators specialize only on the more profitable prey type; (ii) predators are non-adaptive generalists they feed on every encountered prey; and (iii) pre- dators are adaptive generaliststhey behave as optimal foragers. For all three types of behavior we compared the set of parameters that leads to permanence of all three populations. The largest region of parameter space that leads to permanence is for predators that specialize on the more profitable prey type (Fig. 3). This is because, in this case, the alternative prey type not being preyed upon grows logistically and therefore it will always be present. Thus, permanence of a three-dimensional system reduces to permanence of a system which consists of the more profitable prey type and predators only. This model is the RosenzweigMacArthur predatorprey model which has one equilibrium. This equilibrium, if positive (see condi- tion (15)), either is stable or a stable limit cycle exists due to the paradox of enrichment. In both cases predators coexist indefinitely with prey and the model is therefore permanent. If predators behave as generalists and they include the alternative prey type in their diet, this gives more restrictive conditions for permanence of the model (Fig. 2). Such more restrictive conditions are due to the fact that we have to ensure that predators do not drive any of the two prey types to extinction. Holt (1977) showed that by including an alternative prey type in pre- dators' diet the predator equilibrium density increases and the original prey type suffers heavier predation. This leads to lower equilibrium density of the original prey type, or even to exclusion of the original prey type. This indirect effect of one prey type on the other prey species is called predator-mediated apparent competition. The shape of the parameter set that leads to permanence depends on the environmental productivity. If produc- tivity is low, i.e., if the carrying capacity of the more profitable prey type is below the switching threshold, FIG. 4. This figure shows the acyclicity of the boundary equilibria and permanence of system (1) driven by optimal foraging strategy. In then the set of parameters for which the two-preyone 2 1 predator system is permanent is shown in Figs. 2A and all cases E12 is in G .In(A)E13 and E23 exist in G .IfE23 does not exist, 1 2 then we get two cases: E13 exists in G (B) or E13 exists in G (C). 2B. If the carrying capacity for the more profitable prey type is above the switching threshold, then the set of invade when the system consisting of the first prey type parameters leading to permanence is given in Fig. 2C. and predators is at E13 equilibrium because at this equi- If predators follow the optimal diet rule then we show librium predators do not feed on the less profitable prey (Fig. 5) that when the carrying capacity of the more type. This is because as the more profitable prey type profitable prey type is above the switching threshold then becomes abundant, predators exclude the alternative the region of parameter space that leads to permanence prey type from their diet which then recovers until it is for optimal predators is larger than that for generalist included again in the predator diet. predators (Fig. 2C) but smaller than that for specialist File: 653J 139911 . By:SD . Date:01:04:99 . Time:12:48 LOP8M. V8.B. Page 01:01 Codes: 4356 Signs: 3507 . Length: 54 pic 0 pts, 227 mm 122 Kr @van and Sikder FIG. 5. In this figure the region of parameter space that leads to permanence for optimal predators is shown for the same parameters that are used in Fig. 3C. predators (Fig. 3). This is because for predator mortality prey (Ives and Dobson, 1987; Sih, 1987; Ruxton, 1995: rates which are lower than the profitability of the alter- Kr ivan, 1998). For example, Fryxell and Lundberg native prey type the set of parameters leading to per- (1994) assumed a gradual change in profitability of inclu- manence for optimal predators is the same as that for sion of the less profitable prey type. However, this generalist predators while for higher predator mortality gradual change described by a sigmoidal function leads rates the set of parameters leading to permanence for to expressions in permanence analysis which are very dif- optimal foragers is the same as that for specialist ficult to analyze. Our approach, based on the direct use predators. This is due to the fact that for low predator of the step function, allows us to obtain simple conditions mortality rates (m File: 653J 139912 . By:SD . Date:01:04:99 . Time:12:48 LOP8M. V8.B. Page 01:01 Codes: 4504 Signs: 3658 . Length: 54 pic 0 pts, 227 mm Optimal Foraging 123 automatically holds. This condition allowed us to apply and the permanence results of Butler and Waltman and we showed that this theory is a useful tool in some models ma * & 1 2 (c) This case is similar to (b) if we replace E13 by E23 e2 K2 *2 e1 K1 *1+e2 K2 *2 < and E2 by E1 ; see Fig. 1C. 1+h2 K2 *2 1+h1 K1 *1+h2 K2 *2 (d) In this case M=[E0 , E1 , E2 , E12], and the proof is the same as in the previous case; see Fig. 1D. it follows that the condition e K * +e K * m< 1 1 1 2 2 2 , APPENDIX B: PERMANENCE OF (3) 1+h1 K1 *1+h2 K2 *2 Proposition 2. Assume that if a boundary limit cycle which implies that x$3 Âx3 evaluated at E12 is positive, of (3) exists then it repels interior trajectories. Let holds in all cases. Thus, E12 repels interior trajectories. E0 is locally unstable along the x1 and x2 directions but e1 K1 *1 locally stable along the x3 direction. By M we denote the m< (20) set of all feasible boundary equilibria. Since 1+h1 *1 K1 Then system (3) is permanent. x$ e * K 3 = i i i &m, i=1, 2 Proof. Under the given conditions, E is feasible and x3 } Ei 1+hi *i Ki 13 globally asymptotically stable in the plane 62 . Since it follows that if Ei3 is feasible then Ei repels boundary x$ trajectories along the x3 direction. Moreover, E1 always 2 =a2 , repels boundary trajectories in the x2 direction and x2 } E13 similarly for E2 . Thus, all the boundary equilibria are isolated. it follows that E13 repels the interior orbits. Moreover, (a) From inequality (17) it follows that x$ e * K 3 = 1 1 1 &m; x3 } E12 1+h1 *1 K1 M=[E0 , E1 , E2 , E12 , E13 , E23]; i.e., the feasibility of E13 implies that E12 repels interior see Fig. 1A. Since E13 and E23 are feasible, equilibrium Ei trajectories. The remaining part of the proof is the same (i=1, 2) repels orbits locally along the x3 direction. Due as that of part (b) of Proposition 1. to condition (18) both E13 and E23 repel the interior orbits. Thus, the boundary equilibria of (2) have no stable manifold intersecting the interior of the positive octant. From the above analysis of the boundary flow the APPENDIX C: PERMANENCE OF (1) phase portrait for each two-dimensional subsystem reveals the fact that no equilibrium is chained to itself Proposition 3. Assume that if a boundary limit cycle and no subsets of M form a cycle; see Fig. 1A. If repelling of (1) exists then it repels interior trajectories. Optimal Foraging 125 A. Let K1>x*1 . Then system (1) governed by optimal Proposition 3 together with (16) imply that m ma1 *2 & 8 (Ac) This work was supported by GA CR (Grant 201Â98Â0227). The stay of A. 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