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Obligate mutualism with a predator: Stability and persistence of three-species models
THEORETICAL
POPULATION
BIOLOGY
32, 157-175 (1987)
Obligate Mutualism with a Predator: Stability and Persistence of Three-Species Models H. I. FREEDMAN* Department
of Mathematics, The University Canada T6G 2GI
of Alberta,
Edmonton,
JOHN F. ADDICOTT+ Department
of Zoology,
The University of Alberta. Canada T6G 2GI
Edmonton,
AND
BINDHYACHAL Department
of Mathemarics, Allahabad, Uttar
RAI*
The University Pradesh, India
of Allabad,
Received June 17, 1985
We present a general model for three interacting populations, where one population, called a mutualist, benefits a predator in its interaction with the prey. Biologically, there are four different ways in which the mutualist could benefit the predator: by enhancing prey growth rate, by enhancing the rate of prey capture, by providing an alternative food supply for the predator, and by enhancing the efftciency of utilization of prey, once they are ingested. We discuss examples of each type of interaction. We restrict our model to those situations in which the predator cannot survive on the prey in the absence of the mutualist. Therefore, if mutualism exists, it is obligate for the predator. Other conditions of the model include the dynamics of the prey and the mutualist alone and together in the absence of the predator. Given additional reasonable restrictions on the model, we determine the conditions for persistence, where persistence is defined as the continued existence of all three populations without any of them going extinct. There are two ways in which survival may arise in these models. Under one set of conditions, which is equivalent to the predator being able to invade a prey-mutualist system when rare, persistence will occur for any set of positive critical population sizes. Alternatively, survival will occur if there is an asymptotically stable interior equilibrium. * Research partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant NSERC A4823. + Research partially supported by the Natural Sciences and Engineering Research Council of Candada, Grant NSERC A9674. : Research partially supported by U.G.C. Of India, Grant F. g-14/83 (SR-III).
157 0040-5809/87 $3.00 Copyright 0 1987 by Academic Press, Inc. All rights of reproduction in any form reserved.
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However, the conditions for this are complex, and survival may occur only for initial populations in a limited region around the equilibrium. 0 1987 Academic Press, Inc.
Mutualism is an interaction among individuals of two species in which the interaction is beneficial for both (Boucher et al., 1982). Although there are mutualisms which can be modeled as two-species interactions, there are many more in which the dynamics of at least one other species must be considered (Addicott 1984, 1985; Addicott and Freedman 1984; Rai et al., 1983). Individuals of a third species must be present in order for an individual of one species to benefit another. There are a variety of ways in which this can come about. In previous papers on the dynamics of mutualism (Addicott and Freedman, 1984; Freedman et al., 1983; Rai et al., 1983), we modeled interactions among three species leading to mutualism between two of them. We modified existing two-species competitive or predator-prey interactions by adding a third species, called the mutualist. There are three ways in which the mutualist can benefit one but not both species. The mutualist could benefit one of the competitors, the prey, or the predator. The mutualist benefiting the predator in its interaction with the prey has not yet been considered in detail, and we therefore choose to focus on it here. There are four biologically different ways in which a mutualist could benefit the predator: by enhancing prey growth rate, by enhancing rate of prey capture, by providing an alternative food supply to the predator, and by enhancing the efficiency of utilization of prey by predators. First, enhancement of the growth rate of the prey could increase the density of the prey and indirectly increase the rate of prey capture by the predator. Alternatively, increased prey growth rate .could allow greater rates of sustained harvest by the predator and/or decreased impact of the predator on prey density, thereby increasing prey capture over the long run. We cannot find examples of this pattern which are strictly three-species interactions. However, more complex interactions involve this effect. For example, Dodson (1970) describes a system in which salamander larvae are beneficial to an invertebrate predator by feeding the competitors of the prey of the invertebrate predator. Second, the mutualist could provide an alternative food supply to the predator. This would directly affect the growth rate of the predator, facilitating the continued persistence of the predator even in the relative absence of prey. The mutualist could make alternative prey available to the predator, but this would also be a more complex case than the three-
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species system we are considering here. The most likely biological process would be the mutualist producing some form of mutualistic resource such as nectar, honeydew, or food bodies. Although this is a common occurrence in mutualistic systems (Addicott, 1979; Bentley, 1977), it is most common in systems involving benefit to the prey rather than to the predator. Third, a mutualist could increase the probability of successful prey capture by the predator. Any component of the predation process could be affected, including encounter frequency, recognition, pursuit, and attack. The most common biological examples involve situations where the activities of one species cause a change in the habitat use pattern of the prey. This could place the prey into habitats where they would come into greater contact with the predator or where the recognition or pursuit processes would be more successful. For example, stoneflies are known to affect the distributions of mayflies, making mayflies more vulnerable to predation by fish (D. Soluk, personal communciaton). Charnov et al. (1976) discuss two possible examples. Piscivorous marine fish may make prey fish more vulnerable to sea birds by driving the prey towards the surface and army ants may increase the availability of insect prey to ant birds. Thompson et al. (1982) describe the association between cattle egrets and cattle, where the movement of cattle while grazing disturbs insects making them more vulnerable to cattle egrets. Similarly, it is hypothesized that clown fish attract prey to the sea anemones with which the clown fish associate (see Dunn, 1981; Mariscal, 1970; Verwey, 1930). An increase in prey availablity is likely to be exceedingly common, but it is not clear yet how frequently it will occur in mutualistic as opposed to commensal systems. Finally, the mutualist could increase the efftciency with which a predator converts prey biomass into predator biomass. Of the four possible mechanisms, this is probably the best documented. Most of the gut symbioses and endosymbiotic associations of heterotrophs with heterotrophs would fall into this category. For example, the termite-flagellate-bacteria system for the breakdown of cellulose (Breznak, 1975; To et al., 1980) and the microbial floral of ruminants (Hungate, 1966, 1975) are both examples of these kinds of effects. We exclude from analysis some of the possible forms in which this three-species system could exist. For example, the mutualism must be a three-species interaction, with the presence of the prey required. We do not consider situations in which the mutualist and predator could interact on their own in a manner beneficial to each, and where both together could exist in the absence of the prey. We further restrict our analyses to cases in which the predator is obligately dependent upon the mutualist for its survival. We have
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previously analyzed when mutualism is beneficial to the prey population in a viable predator prey system (Addicott and Freedman, 1984), using the following criterion for benefit: LJ(dx/x &)/au > 0 where x is the prey population and u is the mutualist population. It is not clear whether the same kind of definition would carry over to a predator population. If the presence of the mutualist changed the predator prey interaction from one with damped oscillations to one with a stable limit cycle, the (unstable) equilibrium density of the predator could decline and the variance in its _ density could increase. This would not necessarily be “beneficial” to the predator. We have therefore restricted our analyses to cases in which the predator cannot exist on its prey in the absence of the mutualist. Therefore, if mutualism exists,’ it must be obligate mutualism, and it will be easily detected, both in the field and conceptually in our models. We can search for a relatively simple criterion for the existence of mutualism for the predator: Can the predator persist in the presence of the mutualist but not in its absence? Biologically the definition of persistence is relatively straightforward. We consider the predator to be persistent (and therefore the mutualism to be obligate for the predator), if in the absence of stochastic variation the predator population will not decrease to zero. We later apply mathematically rigorous definitions for “strong persistence” and “uniform persistence” (see below). We proceed by presenting our model, including the constraints upon the functions which define the interaction as obligate mutualism. We then rigorously define our concept of persistence and set out the conditions under which all three species would persist. We do this first for the general case, and then for a limited case in which there is an interior equilibrium. THE
MODEL
The following system of automous proposed as a model of a mutualist system (Rai et al., 1983): $=uh(u,
ordinary differential equations is interacting with a predator-prey
x, y)
p = xdu, x) - yp(u, x) $= Y(-s(u) + c(u) P(U,x)1.
(la)
(lb)
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u(t) is the number of mutualists at time t which interact with the prey x(t) and the predator y(t). The meaning and properties of the various components of this model are given below. h(u, x, y) is the specific growth rate of the mutualist. We assume that the following properties hold (Albrecht et al., 1974; Freedman, 1980; Kolmogorov, 1936; Rai et al., 1983). (i)
40, x, Y) > 0,
(ii)
wu, x, Y) Q 0, au
(iii)
ahk X, Y)
ax
d 0,
(iv) amay4 Y)>0, (v)
there exists L(x, y) > 0 such that h(L(x, y), x, y) = 0.
Condition (i) implies that independent of the x and y populations u is capable of growth even when rare and therefore y is a nonobligate mutualist of U. Condition (ii) implies that the specific growth rate of the mutualist is negatively density dependent. Condition (iii) implies that the mutualist does not benefit from the prey, for otherwise we are reduced to a case of two-species mutualism, which is not within the scope of this paper. If the derivative is negative, there will be a cost to the mutualist due to its interactions with the prey. Condition (iv) implies that u derives benefit from the predator population. Condition (v) implies that L(x, y) is the mutualist’s carrying capacity and in part specifies in what way the predator and prey become part of the mutualist’s environment. L(x, y) is a nonincreasing function of x and a nondecreasing function of y. This is because if u helps x (which helps y), this costs u energy which otherwise could be used for growth or reproduction. On the other hand, u derives benefit from y, increasing its growth ability. g(u, x) is the specific growth rate of the prey in the absence of predation. We assume that (vi)
g(u, 0) > 0,
(vii)
-20,
(viii)
ag(u9~)
(ix)
ag(4 -9
au
ax‘
’
there exists K(U) > 0 such that g(u, K(U)) = 0.
(3)
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Condition (vi) simply implies that the prey population is capable of surviving in the presence or absence of the mutualist. Condition (vii) implies that u could increase the growth rate of x in the absence of predation. It is biologically reasonable to assume that the inequality will hold if and only if the inequality holds in condition (iii). Condition (viii) implies increasing intraspecific competition among the prey population as their numbers increase. Condition (ix) implies that K(U), the prey population carrying capacity, may depend on U. K(U) is a nondecreasing function of u and will increase if x derives direct benefit from U. p(u, x) is the predator functional response. The assumptions on p(u, x) are (xl P(& 0) = 0, 64 (xii)
Mu, x) > o au’ ’ @(4
xl
>
(4)
o
--z-’ Condition (x) states that in the absence of prey there can be no predation. Condition (xi) implies that the mutualist could benefit the predator by increasing the rate of successful prey capture. Condition (xii) implies that predation increases as a function of increasing prey density. s(u) is the predator specific death rate in the absence of predation. c(u) is the conversion rate from prey biomass to predator biomass, (xiii) Wu) o --z’ . (xiii) implies that the mutualist could decrease the predator death rate, such as by providing the predator with an alternate food source, and (xiv) is a condition for the mutualist to increase the efficiency of converting prey into predator. As an overriding condition on system (1 ), it is assumed that all functions are sufficiently smooth so that the existence and uniqueness of initial value problems are assured. We have yet to indicate the conditions on system (1) which would make u an obligate mutualist of y. In order to do so, we must consider the predator-prey subsystem of (1) obtained by setting u = 0:
g = MA x) - yp(0,x) (6)
$ = A -s(O) + c(O)l-40,x)).
STABILITY OF THREE-SPECIES MODELS
Such a system has been discussed in Freedman has been shown that if either
163
(1980, Chap. 4). There it
or x*pp,
x*) = so
40)
and
x* 2 K(O),
then the predator population goes extinct; i.e., the predator cannot survive solely on the associated prey in the particular closed environment. For the purposes of this paper we assume that conditions (7) hold. Then if the predator y survives in the presence of U, mutualism exists and is obligate for the predator. In summary, the four possible types of predator mutualism and the conditions on our model which imply them are as follows. The mutualist increasing prey growth is contained in condition (vii). Increasing predation is contained in condition (xi), when ag/au > 0. Making alternative prey available to the predator is contained in condition (xiii) when ds/du < 0. Finally, utilization efficiency is contained in condition (xiv) when dc/du > 0. The concept of persistence itself needs clarification. Biologically, the concept is intuitive; namely, that in cases where all three populations are present, none of them will go extinct. Mathematically we define solutions of system (1) to persist in the sense of Freedman and Waltman (1984); namely, if (u(t), x(t), y(t)) is a solution of system (1) such that u(0) > 0, then lim 1-00 inf u(t) > 0, lim ,,,infx(t)>O, 40) > 0, Y(O) > 0, lim t-02 inf y(t) > 0. This will happen if u helps y sufficiently so that y survives. Such a concept of persistence was referred to as “strong persistence” in Freedman and Waltman (1984). Strong persistence implies that populations may not get “arbitrarily close” to zero as time proceeds. However, how close they do get may depend on the initial population numbers. A better form of persistence would be for populations to be eventually uniformly bounded away from zero, no matter what the initial numbers. Such a concept is termed “uniform persistence.” In Butler et al. (1986) it is shown that for models of the type utilized in this paper, strong persistence implies uniform persistence. In the remainder of this paper we analyze system (1) and give criteria for the existence of uniform persistence.
BOUNDARY
EQUILIBRIA
AND STABILITY
Computation of the boundary equilibria and their stability for system (1) provide the information needed to determine the persistence of the system.
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To do so, we compute the variational (Jacobian) matrix of system (1). The signs of the real parts of the eigenvalues of this matrix evaluated at a given equilibrium determine its stability. This matrix is given by uh,+h vu, 4 Y)’
uk
xgu - YPU
UhY
x&+g-YP,
i Y(-%+cPu+c,P)
-P -s+cp
CYPX
3
(8)
1
where all functions are evaluated at (x, y, z), and where we have used the notation F, to represent aFlax, etc. The simplest equilibrium is E,(O, 0, 0). Here h(O,O, 0) 0
v, = V(0, 0,O) =
[
0
0
0
g(O, 0) 0
0
”
-s(O) I
for which two of the eigenvalues are positive and one is negative, giving a point at the origin with nonempty stable and unstable manifolds (see Fig. 1). Next, the equilibrium I?,(&, 0,O) has variational matrix V,, where L, = L(0, 0). -
L,h,(b,, 0,0) LA&,, v, =
0 0
FIG. arrows
1. Stability are stability
(40) L,h,(L,, ‘A 0)1
g(L
0) 0
0 -s(LJ
of equilibria in the coordinate planes in order to prove directions in the planes. Dashed arrow is perpendicular
persistence. Solid to the U-X plane.
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has two negative eigenvalues and one positive eigenvalue and again has nonempty stable and unstable manifolds (see Fig. 1). Similarly, &(O, K,, 0), where K0 = K(O), and the variational matrix of which is
5 1
v,=
is once more an equilibrium with two negative eigenvalues and one positive eigenvalue (see Fig. 1). There is only one interior planar equilibrium occurring in the U-X plane. Let &ii, 2, 0) be this equilibrium. Then ii, 2 are obtained by solving the algebraic system h(u, x, 0) = g(u, x) = 0, which is equivalent to solving 24= L(x, O),
x = K(u).
(9)
Clearly K(U) is monotonically increasing; i.e., one of the effects of u is to possibly increase the growth of x by increasing its carrying capacity. The various possibilities in solving system (9) are illustrated in Fig. 2. One possibility is that L(x, 0) is constant in x (see Fig. 2). In this case there is always a unique equilibrium ,!?.It is possible that L(x, 0) is a decreasing function of x; there could be a cost to u in helping x grow. If the cost is not
FIG. 2. Unique intersection of x= K(u) and u= L&c, 0). L(x, 0) n is the case where L(x, 0) EL, a constant. L(x, 0) b is the case where L(x, 0) is a decreasing function of x. L(x, 0) c is the case where x = K(u) and u = L(x, 0) fail to intersect. This case does not correspond to mutualism.
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too great (see Fig. 2), then E again exists and is unique. If the cost is so great that u goes extinct by virtue of its carrying capacity becoming zero before intersection of the curves takes place (see Fig. 2), we do not consider this mutualism, and hence we do not allow this case. Thus, there will always be an equilibrium E in the U-X plane in our models (see Fig. 2). The variational matrix about E, P= V(ii, 2, 0) is given by l?h,(ii, 2, 0) r=
-Q,(fi, 2) 0 [
iih,(ii, I, 0) -fg,(k 3 0
iih,(zi, 2,O) -p(ii, 2) . -s(C) + c(E) p(i2, 2) I
The eigenvalues of r are -s(C) + c(C) p(li, 2) and I*, x* = f(iih, + Zg,) f $((iih, + 2gx)* -4Z(h,
(10)
where
g, - h, g,))“‘,
(11)
and all functions are evaluated at (ii, R, 0). Because of the properties described in (2) and (3), h, g, - h, g, > 0 and hence the signs of the real parts of 1, and I- are given by the sign of lh, + Zg,, which is negative. This implies that J!?is asymptotically stable in the U-X plane (see Fig. 1). ,!? is asymptotically stable or unstable in the y direction according to whether -s(C) + c(C) p(ii, 2)
(12)
is negative or positive, respectively. There are no other boundary equilibria. Expression (12) determines whether or not the system (1) persists: if E is asymptotically unstable in the y direction (see Fig. l), then obligate mutualism exists and it is persistent (see below).
PERSISTENCE
Persistence, as defined previously, guarantees the survival of all populations no matter what the initial populations are. One would expect that if the equilibrium E in the U-X plane is unstable in the y direction, then y will persist. Although this may seem intuitive, it is mathematically nontrivial. This result may be stated formally as follows. THEOREM 1. Let the functions defined in system (1) have the properties given by (2), (3), (4), and (5). Further, let expression (12) be positive. Then all solutions of system (1) with positive initial conditions strongly persist, i.e., lim I-00 infu(t)>O, lim,,, inf x(t) > 0, and lim,, m inf y(t) > 0.
The proof of this theorem is given in Appendix 1.
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Once it is known that -s(G) + c(C) p(zi, a) > 0, so that system (1) persists, the remainder of the conditions on system (1) are such that system (1) satisfies the hypotheses of the main theorem in Butler et al. (1986). This implies two further results. The first is that the persistence is uniform: not only do the populations stay away from zero, but they eventually remain at a lixed distance from zero independent of the initial conditions. The second result is that there exists a positive interior equilibrium in (a, x, y) space. This equilibrium is considered following presentations of examples of persistence.
EXAMPLES
OF PERSISTENCE
At this time we present four examples, each illustrating obligate mutualism with a predator in one of the ways we discussed. All examples are of the form u’=au
l(
u
L-Fly-mx
> (13)
In the absence of mutualism, i.e., u=O, there will be a positive equilibrium (x, y) = &/c,y,, (fi/ro)( 1 - s,/c,y,K)), provided c,y,K> s,,. Hence, for the predator not to survive in the absence of mutualism, we require
EXAMPLE 1. m = k = y I = ci = 0, si > 0; i.e., mutualism occurs by means of supplying the predator with the alternate food source. Then ii = L, 2 = K, and the condition for persistence is satisfied if S, > s0 - coyoK/L. EXAMPLE 2. y, = s, = ci = 0, m > 0, k > 0. Mutualism now occurs by means of the mutualist increasing the predator’s primary food source. Here ii = (L - mK)/( 1+ mk), 2 = (kL + K)/( 1 + mk), requiring L > mK. Then the condition for persistence becomes c,y,kL - s,mk > so - coy0 K. EXAMPLE 3. m = k = s1 = c1 = 0, y i > 0. Here the mutualist increases the predator functional response. Again ii = L, 2 = K, and now the condition for persistence reduces to y i > (sO- coy0 K)/c, LK.
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EXAMPLE 4. m = k = y1 = si = 0, ci > 0. This represents mutualism by increasing the conversion factor of biomass from prey to predator. Then ii= L, I = K, and the condition for persistence is satisfied if Cl ’ (&I - %YoWL* STABILITY
OF THE INTERIOR EQUILIBRIUM
We have shown that under the conditions for persistence, there must be an interior equilibrium. However, even if the conditions for persistence do not occur, it is still possible for the predator to survive for certain initial conditions if there exists an asymptotically stable interior equilibrium. An interior equilibrium of the form E*(u*, x*, y*) where U* > 0, x* > 0, y* > 0 may or may not exist. E* will exist if and only if the algebraic system 4% x, Y) = 0 xg(u, x) - H-44 x) = 0
(15)
-s(u) + c(u) p(u, x) = 0 has a positive solution. In Appendix 2 we give algebraic and graphical conditions for E* to exist. If E* exists and I/* E V(u*, x*, y*), then from (8) u*h, v* =
x*&l - Y*Pu
u*h,
u*h,
x*&Y, + g- Y*PX
-P
i Y*(-%!+cP,+cuP)
9
(16)
0 I
CY*PX
where all functions are evaluated at (u*, x*, y*). We define a, = - [u*h,(u*, x*, Y*)
+x*g,(u*, x*) + Au*, x*1 - y*p,(u*, x*11
a2 = u*h,(u*, x*, y*)[x*g,(u*,
x*) + g(u*, x*) - y*p,(u*,
- u*h,(u*, x*, y*)[x*g,(u*, - u*y*h,(u*,
x*)]
x*) - y*p,(u*, x*)]
x*, y*)[ -s,(u*)
+ c(u*) pJu*, x*) + c,(u*) p(u*, x*)]
+ c(u*) Y*P(u*, x*) px(u*, x*) a3 = - c(u*) u*y*p(u*, x*) pJu*, x*) h,(u*, x*, y*) + y*h,(u*, x (x*g,(u*,
x*, y*)[u*(-s,(u*)
+ c(u*)p,(u*,
x*) + c,(u*)p(u*,
x*1 + gtu*, x*) - Y*P,(u*, x*))
- c(u*) u*P,(u*, x*)(x*g,(u*,
x*) - Y*P,(u*,
+ u*y*p(u*, x*) h,(u*, x*, y*)( -s,(u*) + c,(u*) P(u*, x*)).
x*1)1
+ c(u*)) pJu*, x*)
(17) x*))
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Then a,, u2, a3 are the coefficients of the characteristic polynomial of V*. Then utilizing the Routh-Hurwitz criteria (see Sanchez, 1968, p. 57) a necessary and sufficient condition for E* to be asymptotically stable is for a, > 0, a3 > 0, and ala2 - a3 > 0 to hold. An interpretation of the foregoing is that if E* exists and is asymptotically stable, then there is a region near E* within which all solutions are attracted to E*. For these populations, u will be an obligate mutualist of y.
A NUMERICAL
EXAMPLE
To illustrate the analysis of the previous section, the following example is considered:
(18)
In the absence of the mutualist population (U =O), the predator, y, goes extinct exponentially. However, if E= 5, 6 = 0.01, p = 0.1, y = 0.1, ye= 0.01, and c= 560,000/12,221, there is an interior equilibrium given by E*(l, 121/560, 450/56). Checking the stability criterion by evaluating the functions given in (14) gives a, = 67/56 > 0, a3 = 222, 300/6, 283, 816 > 0, u1 u2 - u3 = 474,177,369/3,870,830,656 > 0. Hence E* is asymptotically stable and u is an obligate mutualist of y. Finally, in this case ii = --5 + fi x 0.196, ~=11/2+J27/1o~1.020, s(ii)~0.998, c(fi)p(ii,.Z)~ 0.916, and -s(G) + c(E) p(E, 2) E -0.082 ~0, so that persistence is not satisfied.
DISCUSSION
The results of our analysis can be summarized as follows. First, we have shown that there are conditions under which obligate mutualism can exist between a predator and a mutualist, based upon any of four different ways in which a mutualist could affect the interaction between a predator and its prey. There are at least plausible conditions under which the four types of effects could occur, and for two of them there are well-known biological
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examples. However, this kind of mutualism is poorly known relative to other interactions such as protection mutualisms or direct, pairwise beneficial interactions. Mutualisms involving enhanced conversion ellicienties of ingested prey are well studied (e.g., Hungate, 1966, 1975; Breznak, 1975), but not in an ecological context. Interactions involving enhancement of prey availability are probably extremely common, as the activities of one organism alter the distribution and/or behavior of another (e.g., Charnov et al., 1976). However, these interactions have not received much attention, probably because they arise most frequently as unidirectional, facilitative effects, rather than in the context of mutualism. As all facilitative interactions, whether commensal or mutual, continue to attract the attention of ecologists, particular attention should be placed upon examination of prey-predator-mutualist interactions. They will not be as obvious as protective mutualisms, but they may turn out to be more common. Next, we have shown that obligate mutualism between predator and mutualist will exhibit persistence if a relatively simple set of criteria is met: the predator must be unable to exist on the prey in the absence of the mutualist; both the mutualist and the prey must be capable of persisting in the absence of the other two species; in the absence of the predator, there must be a positive equilibrium (8) for the mutualist and prey. If these conditions hold, then persistence of all three species depends upon the stability characteristics of the mutualist-prey equilibrium, l?, or in more restrictive cases survival depends on the stability of the three-species equilibrium, E*. The three-species system will be uniformly persistent if ,!? is unstable in the direction of predator. In other words, if the predator can invade a community composed of just the mutualist and prey, even when the predator is rare, then all three species will persist. There must be an interior equilibrium, E*, under these conditions, but persistence does not depend at all on the stability properties of the interior equilibrium. The only thing that matters is whether the predator can invade when rare. This requires only that the conversion rate of captured prey times the predator’s functional response be greater than the predator’s intrinsic death rate. Each of these components can potentially be affected by the mutualist, and therefore, the interaction with the mutualist could lead to obligate mutualism. The mutualist could provide alternative food to the predator, increase prey capture rate, or increase conversion of prey biomass into predator biomass. Note however that increasing the growth rate of the prey does not enter into this persistence criterion except as it affects ,??, and therefore could not influence the existence and persistence of mutualism in the same way as the other three effects could. Mutualism is still possible, even if the predator cannot invade when it is rare. However, this requires that an interior equilibrium, E*, exist and that it be asymptotically stable. Other kinds of stability would not suffice. For
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171
example, if E* were unstable and there were some type of periodic orbit around it, its persistence could not be guaranteed. The conditions required to achieve asymptotic stability for E* are exceedingly complex and do not lead to any direct biological interpretations. Previously, we have shown for a variety of two- and three-species models in the sense of local stability of that mutualism can be “stabilizing” equilibria (Addicott and Freedman, 1984; Rai et al., 1983) and return time stability (Addicott, 1981). Our present results show another way that mutualism can be stabilizing in the sense of persistence, providing further confirmation that mutualism will in general, but not always, have a “stability” effect on an existing interaction. The challenge for ecologists now is not just to identify the existence of mutually beneficial interactions but also to begin to examine the dynamics and mechanisms of the regulation of benelkal interactions.
APPENDIX
1
Theorem 1 is proved using techniques developed in Freedman and Waltman (1984). We refer to Fig. 2. In a manner similar to that used in Rai et al. (1983), all solutions of system (1) with nonnegative initial conditions are bounded in positive time. As a consequence, the omega limit set of such solutions is a bounded, closed, connected, and nonempty set. Suppose (u,, x0, y,) is a point in the positive octant, and !G? is the omega limit set of the orbit through (u,,, x0, y,). We will have proved the theorem if we can show that Q does not intersect any of the coordinate planes. If E, ~0, then since &, is a saddle point there is an orbit in the stable manifold of E. belonging to 52 (see Freedman and Waltman, 1984, Appendix 2). But the stable manifold of E, is the y axis which would be that orbit and is unbounded, a contradiction. If E, E Q, since it is a saddle point, it would have an orbit in its stable manifold belonging to Q. E,‘s stable manifold is the u-y plane, and all orbits in this plane are either unbounded or emanate from E,, giving a contradiction. Similarly, E2 x 51. If EE fi, the condition -s(G) + c(G) p(ii, 2) > 0 implies that i? is a saddle point. The stable manifold of ,!? is the U-X plane and hence orbits in this plane emanate from either E,, E,, or E, or are unbounded, once more a contradiction. Since no planar equilibrium is in the omega limit set, 8, no planar point is in Q since there are no other closed orbits in the planes. This proves the theorem.
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AND
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2
In order for E* to exist, the algebraic system Mu, 4 Y) = 0
UW
xg(u, x) - J&u, x) = 0
(15b)
-s(u) + c(u) p(u, x) = 0
(15c)
must have a positive solution. We first consider the curve given by (1%). From (4) and (5) this is a montonically decreasing curve in the u--x plane (see Fig. 3) and hence defines x as a function of U, x = q(u), such that either (I) there exists u0 > 0 such that lim, _ U; q(u) = +co (Fig. 3, case I), or (II) ~(0) =,? > 0 (Fig. 3, case II). In both cases dq/du
Y)
(18a)
r*: Y = f-P(u) A4 cp(U)YP(4 cp(u)).
(18b)
As far as ri is concerned, since dL/au = (aL/ax)(&p/au) 3 0, as u varies from zero to infinity, L(rp(u), u) for fixed y varies from L(m, y) to
CASE
MUTUALIST
II
(u)
FIG. 3. Case I: curve in the u-x plane defining q(u), determined by -s(u) where lim,,,+ q(u) = fco. Case II: curve in the u-x plane defining q(u), -s(u) + c(u) i(u, x) = 0, where lim,,,+ q(u) = P.
+ cp(u, x) = 0, determined by
STABILITY OF THREE-SPECIES MODELS
MUTUALIST
173
(u)
FIG. 4. r1 in case II when (a) aLjay < 0, (b) aL/@ = 0
L(q,, y) as u varies from q, to co in case I and L(& y) to L(cp,, y) in case II, where cpm= lim, _ m q(u). Hence, for fixed y, (18a) always has a solution u(y) in case II, but may or may not have such a solution in case I. If such a solution exists, since aLjay ~0, as y is increased, u is not increased for a solution (i.e., dy/du = (1 - aL/au)/(iTL/ay) < 0 if al;/@ < 0). We indicate f 1 in case II in Fig. 4, where on Z, , u = ri when y = 0. We now discuss Z2 restricting ourselves to case II. Utilizing (lk), (18b) may be written as
y = c(u) du) g(u, du)) s(u)
.
(19)
At u =O, y,,= c(0) 2g(O, 2)/s(O). y, > 0 if and only if 2 0 on Z,, E* may not exist. Hence in order for E* to exist we must have that Tz is an increasing function of y and that Z, crosses the u axis at u = ii, where U-C ~2.Otherwise E* will not exist (see Fig. 5).
MUTUALIST
(u)
FIG. 5. Possible curves r,. Only in case (d) can E* exist.
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AND RAI
In case I, the analysis is even more complicated understanding of the problem.
and adds nothing to our
REFERENCES J. F. 1981. Stability properties of 2-species models of mutualism: Simulation studies, Oecologia 49, 42-49. ADDICOTT, J. F. 1984. Mutualistic interactions in population and community processes, in “A New Ecology: Novel Approaches to Interactive Systems” (P. W. Price, C. N. Slobodchikoff, and B. S. Gaud, Eds.) pp. 437455, Wiley, New York. ADDICOIT, J. F. 1985. On the population consequences of mutuahsm. “Community Ecology” (T. Case and J. Diamond, Eds.), pp. 425-436, Univ. of California Press, Berkeley. ADDICOTT, J. F., AND FREEDMAN, H. I. 1984. On the structure and stability of mutualistic systems: Analysis of predator-prey and competition models as modified by the action of a slow growing mutualist, Theor. Popul. Biol. 26, 32&339. ALBRECHT, F., GATZKE, H., HADDAD, A., AND WAX, N. 1974. The dynamics of two interacting populations, J. Math. Anal. Appl. 46, 658-670. BENTLEY, B. L. 1977. Extrafloral nectaries and protection by pugnacious bodyguards, ADDICOTT,
Annu. Rev. Ecol. Syst. 8, 407427. BOUCHER,
D. H., JAMES, S., AND KEELER, K. H. 1982. The ecology of mutualism, Annu. Rev. 315-347. J. A. 1975. Symbiotic relationships between termites and their intestinal microbiota,
Ecol. Syst. 13, BREZNAK,
Symp.
Sot. Exp.
Biol. 29, 559-580.
BUTLER, G. J., FREEDMAN,H. I., AND WALTMAN, P. 1986. Uniformly persistent systems, Proc. AMS
96, 425-430.
DEAN, A. M. 1983. A simple model of mutualism, Amer. Nat. 121, 409417. DODSON, S. I. 1970. Complementary feeding niches sustained by size-selective predation, Limnol. Oceanogr. 15, 131-137. DUNN, D. F. 1981. The clownfish sea anamones: Stichodactylidae (Coelenterata: Actinaria) and other sea anemones symbiotic with pomacentrid fishes, Trans. Amer. Philos. Sot. 71, 1-115. FREEDMAN,H. I. 1980. “Deterministic Mathematical Models in Population Ecology,” Dekker, New York. FREEDMAN,H. I., ADDICOTT, J. F., AND RAI, B. 1983. Nonobligate and obligate models of mutualism, in “Population Biology Proceedings, Edmonton 1982” (H. I. Freedman and C. Strobeck, Eds.), pp. 349-354, Springer-Verlag, Heidelberg. FREEDMAN, H. I., AND WALTMAN, P. 1984. Persistence in models in three interacting predator-prey populations, Math. Biosci. 68, 213-231. FREEDMAN, H. I., AND WALTMAN, P. 1985. Persistence in a model of three competitive populations, Math. Biosci. 73, 89-101. HUNGATE, R. E. 1966. “The Rumen and its Microbes,” Academic Press, New York. HUNGATE, R. E. 1975. The rumen microbial ecosystem, Annu. Rev. Ecol. Syst. 6, 39-66. KOLM~GOROV, A. N. 1936. Sulla teoria di Volterra della lotta per l’esisttenza, Gior. Zsrirulo Ital. Attuari
7, 74-80.
MARISCAL, R. N. 1970. The nature of the symbiosis between Indo-Pacific anemone fishes and sea anemones, Mar. Biol. 6:58-65. MAY, R. M. 1976. Models for two interacting populations, in “Theoretical Ecology” (R. M. May, Ed.), pp. 49-70, Saunders, Philadelphia. RAI, B., FREEDMAN, H. I., AND ADDICOTT, J. F. 1983. Analysis of three species models of mutualism in predator-prey and competitive systems, Math. Biosci. 63, lr50.
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SANCHEZ,D. A. 1968. “Ordinary Differential Equations,” Freeman, San Fransisco. THOMPSON, C. F., LANYON, S. M., AND THOMPSON, K. M. 1982. The influence of foraging benefits on association of cattle egrets (Bubulcus ibis) with cattle, Oecologiu 52, 167-170. To, L. P., MARGULIS, L., AND NUTTING, W. L. 1980. The symbiotic microbial community of the Sonoran desert termite: Pterotermes occidentis, Biosysfems 13, 109-137. VANDERMEER,J. H. 1980. Indirect mutualism: Varieties on a theme by Stephen Levine, Amer. Nat.
116, 441448.
VANDERMEER, J. H., AND BOXHER, D. H. 1978. Varieties of mutualistic interaction in population models. J. Theor. Biol. 74, 549-558. VERWEY,J. 1930. Coral reef studies. 1. The symbiosis between damsel fishes and sea anemones in Batavia Bay, Treubiu 12, 305-366.
POPULATION
BIOLOGY
32, 157-175 (1987)
Obligate Mutualism with a Predator: Stability and Persistence of Three-Species Models H. I. FREEDMAN* Department
of Mathematics, The University Canada T6G 2GI
of Alberta,
Edmonton,
JOHN F. ADDICOTT+ Department
of Zoology,
The University of Alberta. Canada T6G 2GI
Edmonton,
AND
BINDHYACHAL Department
of Mathemarics, Allahabad, Uttar
RAI*
The University Pradesh, India
of Allabad,
Received June 17, 1985
We present a general model for three interacting populations, where one population, called a mutualist, benefits a predator in its interaction with the prey. Biologically, there are four different ways in which the mutualist could benefit the predator: by enhancing prey growth rate, by enhancing the rate of prey capture, by providing an alternative food supply for the predator, and by enhancing the efftciency of utilization of prey, once they are ingested. We discuss examples of each type of interaction. We restrict our model to those situations in which the predator cannot survive on the prey in the absence of the mutualist. Therefore, if mutualism exists, it is obligate for the predator. Other conditions of the model include the dynamics of the prey and the mutualist alone and together in the absence of the predator. Given additional reasonable restrictions on the model, we determine the conditions for persistence, where persistence is defined as the continued existence of all three populations without any of them going extinct. There are two ways in which survival may arise in these models. Under one set of conditions, which is equivalent to the predator being able to invade a prey-mutualist system when rare, persistence will occur for any set of positive critical population sizes. Alternatively, survival will occur if there is an asymptotically stable interior equilibrium. * Research partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant NSERC A4823. + Research partially supported by the Natural Sciences and Engineering Research Council of Candada, Grant NSERC A9674. : Research partially supported by U.G.C. Of India, Grant F. g-14/83 (SR-III).
157 0040-5809/87 $3.00 Copyright 0 1987 by Academic Press, Inc. All rights of reproduction in any form reserved.
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However, the conditions for this are complex, and survival may occur only for initial populations in a limited region around the equilibrium. 0 1987 Academic Press, Inc.
Mutualism is an interaction among individuals of two species in which the interaction is beneficial for both (Boucher et al., 1982). Although there are mutualisms which can be modeled as two-species interactions, there are many more in which the dynamics of at least one other species must be considered (Addicott 1984, 1985; Addicott and Freedman 1984; Rai et al., 1983). Individuals of a third species must be present in order for an individual of one species to benefit another. There are a variety of ways in which this can come about. In previous papers on the dynamics of mutualism (Addicott and Freedman, 1984; Freedman et al., 1983; Rai et al., 1983), we modeled interactions among three species leading to mutualism between two of them. We modified existing two-species competitive or predator-prey interactions by adding a third species, called the mutualist. There are three ways in which the mutualist can benefit one but not both species. The mutualist could benefit one of the competitors, the prey, or the predator. The mutualist benefiting the predator in its interaction with the prey has not yet been considered in detail, and we therefore choose to focus on it here. There are four biologically different ways in which a mutualist could benefit the predator: by enhancing prey growth rate, by enhancing rate of prey capture, by providing an alternative food supply to the predator, and by enhancing the efficiency of utilization of prey by predators. First, enhancement of the growth rate of the prey could increase the density of the prey and indirectly increase the rate of prey capture by the predator. Alternatively, increased prey growth rate .could allow greater rates of sustained harvest by the predator and/or decreased impact of the predator on prey density, thereby increasing prey capture over the long run. We cannot find examples of this pattern which are strictly three-species interactions. However, more complex interactions involve this effect. For example, Dodson (1970) describes a system in which salamander larvae are beneficial to an invertebrate predator by feeding the competitors of the prey of the invertebrate predator. Second, the mutualist could provide an alternative food supply to the predator. This would directly affect the growth rate of the predator, facilitating the continued persistence of the predator even in the relative absence of prey. The mutualist could make alternative prey available to the predator, but this would also be a more complex case than the three-
STABILITY
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159
species system we are considering here. The most likely biological process would be the mutualist producing some form of mutualistic resource such as nectar, honeydew, or food bodies. Although this is a common occurrence in mutualistic systems (Addicott, 1979; Bentley, 1977), it is most common in systems involving benefit to the prey rather than to the predator. Third, a mutualist could increase the probability of successful prey capture by the predator. Any component of the predation process could be affected, including encounter frequency, recognition, pursuit, and attack. The most common biological examples involve situations where the activities of one species cause a change in the habitat use pattern of the prey. This could place the prey into habitats where they would come into greater contact with the predator or where the recognition or pursuit processes would be more successful. For example, stoneflies are known to affect the distributions of mayflies, making mayflies more vulnerable to predation by fish (D. Soluk, personal communciaton). Charnov et al. (1976) discuss two possible examples. Piscivorous marine fish may make prey fish more vulnerable to sea birds by driving the prey towards the surface and army ants may increase the availability of insect prey to ant birds. Thompson et al. (1982) describe the association between cattle egrets and cattle, where the movement of cattle while grazing disturbs insects making them more vulnerable to cattle egrets. Similarly, it is hypothesized that clown fish attract prey to the sea anemones with which the clown fish associate (see Dunn, 1981; Mariscal, 1970; Verwey, 1930). An increase in prey availablity is likely to be exceedingly common, but it is not clear yet how frequently it will occur in mutualistic as opposed to commensal systems. Finally, the mutualist could increase the efftciency with which a predator converts prey biomass into predator biomass. Of the four possible mechanisms, this is probably the best documented. Most of the gut symbioses and endosymbiotic associations of heterotrophs with heterotrophs would fall into this category. For example, the termite-flagellate-bacteria system for the breakdown of cellulose (Breznak, 1975; To et al., 1980) and the microbial floral of ruminants (Hungate, 1966, 1975) are both examples of these kinds of effects. We exclude from analysis some of the possible forms in which this three-species system could exist. For example, the mutualism must be a three-species interaction, with the presence of the prey required. We do not consider situations in which the mutualist and predator could interact on their own in a manner beneficial to each, and where both together could exist in the absence of the prey. We further restrict our analyses to cases in which the predator is obligately dependent upon the mutualist for its survival. We have
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previously analyzed when mutualism is beneficial to the prey population in a viable predator prey system (Addicott and Freedman, 1984), using the following criterion for benefit: LJ(dx/x &)/au > 0 where x is the prey population and u is the mutualist population. It is not clear whether the same kind of definition would carry over to a predator population. If the presence of the mutualist changed the predator prey interaction from one with damped oscillations to one with a stable limit cycle, the (unstable) equilibrium density of the predator could decline and the variance in its _ density could increase. This would not necessarily be “beneficial” to the predator. We have therefore restricted our analyses to cases in which the predator cannot exist on its prey in the absence of the mutualist. Therefore, if mutualism exists,’ it must be obligate mutualism, and it will be easily detected, both in the field and conceptually in our models. We can search for a relatively simple criterion for the existence of mutualism for the predator: Can the predator persist in the presence of the mutualist but not in its absence? Biologically the definition of persistence is relatively straightforward. We consider the predator to be persistent (and therefore the mutualism to be obligate for the predator), if in the absence of stochastic variation the predator population will not decrease to zero. We later apply mathematically rigorous definitions for “strong persistence” and “uniform persistence” (see below). We proceed by presenting our model, including the constraints upon the functions which define the interaction as obligate mutualism. We then rigorously define our concept of persistence and set out the conditions under which all three species would persist. We do this first for the general case, and then for a limited case in which there is an interior equilibrium. THE
MODEL
The following system of automous proposed as a model of a mutualist system (Rai et al., 1983): $=uh(u,
ordinary differential equations is interacting with a predator-prey
x, y)
p = xdu, x) - yp(u, x) $= Y(-s(u) + c(u) P(U,x)1.
(la)
(lb)
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MODELS
161
u(t) is the number of mutualists at time t which interact with the prey x(t) and the predator y(t). The meaning and properties of the various components of this model are given below. h(u, x, y) is the specific growth rate of the mutualist. We assume that the following properties hold (Albrecht et al., 1974; Freedman, 1980; Kolmogorov, 1936; Rai et al., 1983). (i)
40, x, Y) > 0,
(ii)
wu, x, Y) Q 0, au
(iii)
ahk X, Y)
ax
d 0,
(iv) amay4 Y)>0, (v)
there exists L(x, y) > 0 such that h(L(x, y), x, y) = 0.
Condition (i) implies that independent of the x and y populations u is capable of growth even when rare and therefore y is a nonobligate mutualist of U. Condition (ii) implies that the specific growth rate of the mutualist is negatively density dependent. Condition (iii) implies that the mutualist does not benefit from the prey, for otherwise we are reduced to a case of two-species mutualism, which is not within the scope of this paper. If the derivative is negative, there will be a cost to the mutualist due to its interactions with the prey. Condition (iv) implies that u derives benefit from the predator population. Condition (v) implies that L(x, y) is the mutualist’s carrying capacity and in part specifies in what way the predator and prey become part of the mutualist’s environment. L(x, y) is a nonincreasing function of x and a nondecreasing function of y. This is because if u helps x (which helps y), this costs u energy which otherwise could be used for growth or reproduction. On the other hand, u derives benefit from y, increasing its growth ability. g(u, x) is the specific growth rate of the prey in the absence of predation. We assume that (vi)
g(u, 0) > 0,
(vii)
-20,
(viii)
ag(u9~)
(ix)
ag(4 -9
au
ax‘
’
there exists K(U) > 0 such that g(u, K(U)) = 0.
(3)
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Condition (vi) simply implies that the prey population is capable of surviving in the presence or absence of the mutualist. Condition (vii) implies that u could increase the growth rate of x in the absence of predation. It is biologically reasonable to assume that the inequality will hold if and only if the inequality holds in condition (iii). Condition (viii) implies increasing intraspecific competition among the prey population as their numbers increase. Condition (ix) implies that K(U), the prey population carrying capacity, may depend on U. K(U) is a nondecreasing function of u and will increase if x derives direct benefit from U. p(u, x) is the predator functional response. The assumptions on p(u, x) are (xl P(& 0) = 0, 64 (xii)
Mu, x) > o au’ ’ @(4
xl
>
(4)
o
--z-’ Condition (x) states that in the absence of prey there can be no predation. Condition (xi) implies that the mutualist could benefit the predator by increasing the rate of successful prey capture. Condition (xii) implies that predation increases as a function of increasing prey density. s(u) is the predator specific death rate in the absence of predation. c(u) is the conversion rate from prey biomass to predator biomass, (xiii) Wu)
g = MA x) - yp(0,x) (6)
$ = A -s(O) + c(O)l-40,x)).
STABILITY OF THREE-SPECIES MODELS
Such a system has been discussed in Freedman has been shown that if either
163
(1980, Chap. 4). There it
or x*pp,
x*) = so
40)
and
x* 2 K(O),
then the predator population goes extinct; i.e., the predator cannot survive solely on the associated prey in the particular closed environment. For the purposes of this paper we assume that conditions (7) hold. Then if the predator y survives in the presence of U, mutualism exists and is obligate for the predator. In summary, the four possible types of predator mutualism and the conditions on our model which imply them are as follows. The mutualist increasing prey growth is contained in condition (vii). Increasing predation is contained in condition (xi), when ag/au > 0. Making alternative prey available to the predator is contained in condition (xiii) when ds/du < 0. Finally, utilization efficiency is contained in condition (xiv) when dc/du > 0. The concept of persistence itself needs clarification. Biologically, the concept is intuitive; namely, that in cases where all three populations are present, none of them will go extinct. Mathematically we define solutions of system (1) to persist in the sense of Freedman and Waltman (1984); namely, if (u(t), x(t), y(t)) is a solution of system (1) such that u(0) > 0, then lim 1-00 inf u(t) > 0, lim ,,,infx(t)>O, 40) > 0, Y(O) > 0, lim t-02 inf y(t) > 0. This will happen if u helps y sufficiently so that y survives. Such a concept of persistence was referred to as “strong persistence” in Freedman and Waltman (1984). Strong persistence implies that populations may not get “arbitrarily close” to zero as time proceeds. However, how close they do get may depend on the initial population numbers. A better form of persistence would be for populations to be eventually uniformly bounded away from zero, no matter what the initial numbers. Such a concept is termed “uniform persistence.” In Butler et al. (1986) it is shown that for models of the type utilized in this paper, strong persistence implies uniform persistence. In the remainder of this paper we analyze system (1) and give criteria for the existence of uniform persistence.
BOUNDARY
EQUILIBRIA
AND STABILITY
Computation of the boundary equilibria and their stability for system (1) provide the information needed to determine the persistence of the system.
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To do so, we compute the variational (Jacobian) matrix of system (1). The signs of the real parts of the eigenvalues of this matrix evaluated at a given equilibrium determine its stability. This matrix is given by uh,+h vu, 4 Y)’
uk
xgu - YPU
UhY
x&+g-YP,
i Y(-%+cPu+c,P)
-P -s+cp
CYPX
3
(8)
1
where all functions are evaluated at (x, y, z), and where we have used the notation F, to represent aFlax, etc. The simplest equilibrium is E,(O, 0, 0). Here h(O,O, 0) 0
v, = V(0, 0,O) =
[
0
0
0
g(O, 0) 0
0
”
-s(O) I
for which two of the eigenvalues are positive and one is negative, giving a point at the origin with nonempty stable and unstable manifolds (see Fig. 1). Next, the equilibrium I?,(&, 0,O) has variational matrix V,, where L, = L(0, 0). -
L,h,(b,, 0,0) LA&,, v, =
0 0
FIG. arrows
1. Stability are stability
(40) L,h,(L,, ‘A 0)1
g(L
0) 0
0 -s(LJ
of equilibria in the coordinate planes in order to prove directions in the planes. Dashed arrow is perpendicular
persistence. Solid to the U-X plane.
STABILITY
OF THREE-SPECIES MODELS
165
has two negative eigenvalues and one positive eigenvalue and again has nonempty stable and unstable manifolds (see Fig. 1). Similarly, &(O, K,, 0), where K0 = K(O), and the variational matrix of which is
5 1
v,=
is once more an equilibrium with two negative eigenvalues and one positive eigenvalue (see Fig. 1). There is only one interior planar equilibrium occurring in the U-X plane. Let &ii, 2, 0) be this equilibrium. Then ii, 2 are obtained by solving the algebraic system h(u, x, 0) = g(u, x) = 0, which is equivalent to solving 24= L(x, O),
x = K(u).
(9)
Clearly K(U) is monotonically increasing; i.e., one of the effects of u is to possibly increase the growth of x by increasing its carrying capacity. The various possibilities in solving system (9) are illustrated in Fig. 2. One possibility is that L(x, 0) is constant in x (see Fig. 2). In this case there is always a unique equilibrium ,!?.It is possible that L(x, 0) is a decreasing function of x; there could be a cost to u in helping x grow. If the cost is not
FIG. 2. Unique intersection of x= K(u) and u= L&c, 0). L(x, 0) n is the case where L(x, 0) EL, a constant. L(x, 0) b is the case where L(x, 0) is a decreasing function of x. L(x, 0) c is the case where x = K(u) and u = L(x, 0) fail to intersect. This case does not correspond to mutualism.
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too great (see Fig. 2), then E again exists and is unique. If the cost is so great that u goes extinct by virtue of its carrying capacity becoming zero before intersection of the curves takes place (see Fig. 2), we do not consider this mutualism, and hence we do not allow this case. Thus, there will always be an equilibrium E in the U-X plane in our models (see Fig. 2). The variational matrix about E, P= V(ii, 2, 0) is given by l?h,(ii, 2, 0) r=
-Q,(fi, 2) 0 [
iih,(ii, I, 0) -fg,(k 3 0
iih,(zi, 2,O) -p(ii, 2) . -s(C) + c(E) p(i2, 2) I
The eigenvalues of r are -s(C) + c(C) p(li, 2) and I*, x* = f(iih, + Zg,) f $((iih, + 2gx)* -4Z(h,
(10)
where
g, - h, g,))“‘,
(11)
and all functions are evaluated at (ii, R, 0). Because of the properties described in (2) and (3), h, g, - h, g, > 0 and hence the signs of the real parts of 1, and I- are given by the sign of lh, + Zg,, which is negative. This implies that J!?is asymptotically stable in the U-X plane (see Fig. 1). ,!? is asymptotically stable or unstable in the y direction according to whether -s(C) + c(C) p(ii, 2)
(12)
is negative or positive, respectively. There are no other boundary equilibria. Expression (12) determines whether or not the system (1) persists: if E is asymptotically unstable in the y direction (see Fig. l), then obligate mutualism exists and it is persistent (see below).
PERSISTENCE
Persistence, as defined previously, guarantees the survival of all populations no matter what the initial populations are. One would expect that if the equilibrium E in the U-X plane is unstable in the y direction, then y will persist. Although this may seem intuitive, it is mathematically nontrivial. This result may be stated formally as follows. THEOREM 1. Let the functions defined in system (1) have the properties given by (2), (3), (4), and (5). Further, let expression (12) be positive. Then all solutions of system (1) with positive initial conditions strongly persist, i.e., lim I-00 infu(t)>O, lim,,, inf x(t) > 0, and lim,, m inf y(t) > 0.
The proof of this theorem is given in Appendix 1.
STABILITYOFTHREE-SPECIES
MODELS
167
Once it is known that -s(G) + c(C) p(zi, a) > 0, so that system (1) persists, the remainder of the conditions on system (1) are such that system (1) satisfies the hypotheses of the main theorem in Butler et al. (1986). This implies two further results. The first is that the persistence is uniform: not only do the populations stay away from zero, but they eventually remain at a lixed distance from zero independent of the initial conditions. The second result is that there exists a positive interior equilibrium in (a, x, y) space. This equilibrium is considered following presentations of examples of persistence.
EXAMPLES
OF PERSISTENCE
At this time we present four examples, each illustrating obligate mutualism with a predator in one of the ways we discussed. All examples are of the form u’=au
l(
u
L-Fly-mx
> (13)
In the absence of mutualism, i.e., u=O, there will be a positive equilibrium (x, y) = &/c,y,, (fi/ro)( 1 - s,/c,y,K)), provided c,y,K> s,,. Hence, for the predator not to survive in the absence of mutualism, we require
EXAMPLE 1. m = k = y I = ci = 0, si > 0; i.e., mutualism occurs by means of supplying the predator with the alternate food source. Then ii = L, 2 = K, and the condition for persistence is satisfied if S, > s0 - coyoK/L. EXAMPLE 2. y, = s, = ci = 0, m > 0, k > 0. Mutualism now occurs by means of the mutualist increasing the predator’s primary food source. Here ii = (L - mK)/( 1+ mk), 2 = (kL + K)/( 1 + mk), requiring L > mK. Then the condition for persistence becomes c,y,kL - s,mk > so - coy0 K. EXAMPLE 3. m = k = s1 = c1 = 0, y i > 0. Here the mutualist increases the predator functional response. Again ii = L, 2 = K, and now the condition for persistence reduces to y i > (sO- coy0 K)/c, LK.
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EXAMPLE 4. m = k = y1 = si = 0, ci > 0. This represents mutualism by increasing the conversion factor of biomass from prey to predator. Then ii= L, I = K, and the condition for persistence is satisfied if Cl ’ (&I - %YoWL* STABILITY
OF THE INTERIOR EQUILIBRIUM
We have shown that under the conditions for persistence, there must be an interior equilibrium. However, even if the conditions for persistence do not occur, it is still possible for the predator to survive for certain initial conditions if there exists an asymptotically stable interior equilibrium. An interior equilibrium of the form E*(u*, x*, y*) where U* > 0, x* > 0, y* > 0 may or may not exist. E* will exist if and only if the algebraic system 4% x, Y) = 0 xg(u, x) - H-44 x) = 0
(15)
-s(u) + c(u) p(u, x) = 0 has a positive solution. In Appendix 2 we give algebraic and graphical conditions for E* to exist. If E* exists and I/* E V(u*, x*, y*), then from (8) u*h, v* =
x*&l - Y*Pu
u*h,
u*h,
x*&Y, + g- Y*PX
-P
i Y*(-%!+cP,+cuP)
9
(16)
0 I
CY*PX
where all functions are evaluated at (u*, x*, y*). We define a, = - [u*h,(u*, x*, Y*)
+x*g,(u*, x*) + Au*, x*1 - y*p,(u*, x*11
a2 = u*h,(u*, x*, y*)[x*g,(u*,
x*) + g(u*, x*) - y*p,(u*,
- u*h,(u*, x*, y*)[x*g,(u*, - u*y*h,(u*,
x*)]
x*) - y*p,(u*, x*)]
x*, y*)[ -s,(u*)
+ c(u*) pJu*, x*) + c,(u*) p(u*, x*)]
+ c(u*) Y*P(u*, x*) px(u*, x*) a3 = - c(u*) u*y*p(u*, x*) pJu*, x*) h,(u*, x*, y*) + y*h,(u*, x (x*g,(u*,
x*, y*)[u*(-s,(u*)
+ c(u*)p,(u*,
x*) + c,(u*)p(u*,
x*1 + gtu*, x*) - Y*P,(u*, x*))
- c(u*) u*P,(u*, x*)(x*g,(u*,
x*) - Y*P,(u*,
+ u*y*p(u*, x*) h,(u*, x*, y*)( -s,(u*) + c,(u*) P(u*, x*)).
x*1)1
+ c(u*)) pJu*, x*)
(17) x*))
STABILITY OF THREE-SPECIES MODELS
169
Then a,, u2, a3 are the coefficients of the characteristic polynomial of V*. Then utilizing the Routh-Hurwitz criteria (see Sanchez, 1968, p. 57) a necessary and sufficient condition for E* to be asymptotically stable is for a, > 0, a3 > 0, and ala2 - a3 > 0 to hold. An interpretation of the foregoing is that if E* exists and is asymptotically stable, then there is a region near E* within which all solutions are attracted to E*. For these populations, u will be an obligate mutualist of y.
A NUMERICAL
EXAMPLE
To illustrate the analysis of the previous section, the following example is considered:
(18)
In the absence of the mutualist population (U =O), the predator, y, goes extinct exponentially. However, if E= 5, 6 = 0.01, p = 0.1, y = 0.1, ye= 0.01, and c= 560,000/12,221, there is an interior equilibrium given by E*(l, 121/560, 450/56). Checking the stability criterion by evaluating the functions given in (14) gives a, = 67/56 > 0, a3 = 222, 300/6, 283, 816 > 0, u1 u2 - u3 = 474,177,369/3,870,830,656 > 0. Hence E* is asymptotically stable and u is an obligate mutualist of y. Finally, in this case ii = --5 + fi x 0.196, ~=11/2+J27/1o~1.020, s(ii)~0.998, c(fi)p(ii,.Z)~ 0.916, and -s(G) + c(E) p(E, 2) E -0.082 ~0, so that persistence is not satisfied.
DISCUSSION
The results of our analysis can be summarized as follows. First, we have shown that there are conditions under which obligate mutualism can exist between a predator and a mutualist, based upon any of four different ways in which a mutualist could affect the interaction between a predator and its prey. There are at least plausible conditions under which the four types of effects could occur, and for two of them there are well-known biological
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examples. However, this kind of mutualism is poorly known relative to other interactions such as protection mutualisms or direct, pairwise beneficial interactions. Mutualisms involving enhanced conversion ellicienties of ingested prey are well studied (e.g., Hungate, 1966, 1975; Breznak, 1975), but not in an ecological context. Interactions involving enhancement of prey availability are probably extremely common, as the activities of one organism alter the distribution and/or behavior of another (e.g., Charnov et al., 1976). However, these interactions have not received much attention, probably because they arise most frequently as unidirectional, facilitative effects, rather than in the context of mutualism. As all facilitative interactions, whether commensal or mutual, continue to attract the attention of ecologists, particular attention should be placed upon examination of prey-predator-mutualist interactions. They will not be as obvious as protective mutualisms, but they may turn out to be more common. Next, we have shown that obligate mutualism between predator and mutualist will exhibit persistence if a relatively simple set of criteria is met: the predator must be unable to exist on the prey in the absence of the mutualist; both the mutualist and the prey must be capable of persisting in the absence of the other two species; in the absence of the predator, there must be a positive equilibrium (8) for the mutualist and prey. If these conditions hold, then persistence of all three species depends upon the stability characteristics of the mutualist-prey equilibrium, l?, or in more restrictive cases survival depends on the stability of the three-species equilibrium, E*. The three-species system will be uniformly persistent if ,!? is unstable in the direction of predator. In other words, if the predator can invade a community composed of just the mutualist and prey, even when the predator is rare, then all three species will persist. There must be an interior equilibrium, E*, under these conditions, but persistence does not depend at all on the stability properties of the interior equilibrium. The only thing that matters is whether the predator can invade when rare. This requires only that the conversion rate of captured prey times the predator’s functional response be greater than the predator’s intrinsic death rate. Each of these components can potentially be affected by the mutualist, and therefore, the interaction with the mutualist could lead to obligate mutualism. The mutualist could provide alternative food to the predator, increase prey capture rate, or increase conversion of prey biomass into predator biomass. Note however that increasing the growth rate of the prey does not enter into this persistence criterion except as it affects ,??, and therefore could not influence the existence and persistence of mutualism in the same way as the other three effects could. Mutualism is still possible, even if the predator cannot invade when it is rare. However, this requires that an interior equilibrium, E*, exist and that it be asymptotically stable. Other kinds of stability would not suffice. For
STABILITY
OF THREE-SPECIES
MODELS
171
example, if E* were unstable and there were some type of periodic orbit around it, its persistence could not be guaranteed. The conditions required to achieve asymptotic stability for E* are exceedingly complex and do not lead to any direct biological interpretations. Previously, we have shown for a variety of two- and three-species models in the sense of local stability of that mutualism can be “stabilizing” equilibria (Addicott and Freedman, 1984; Rai et al., 1983) and return time stability (Addicott, 1981). Our present results show another way that mutualism can be stabilizing in the sense of persistence, providing further confirmation that mutualism will in general, but not always, have a “stability” effect on an existing interaction. The challenge for ecologists now is not just to identify the existence of mutually beneficial interactions but also to begin to examine the dynamics and mechanisms of the regulation of benelkal interactions.
APPENDIX
1
Theorem 1 is proved using techniques developed in Freedman and Waltman (1984). We refer to Fig. 2. In a manner similar to that used in Rai et al. (1983), all solutions of system (1) with nonnegative initial conditions are bounded in positive time. As a consequence, the omega limit set of such solutions is a bounded, closed, connected, and nonempty set. Suppose (u,, x0, y,) is a point in the positive octant, and !G? is the omega limit set of the orbit through (u,,, x0, y,). We will have proved the theorem if we can show that Q does not intersect any of the coordinate planes. If E, ~0, then since &, is a saddle point there is an orbit in the stable manifold of E. belonging to 52 (see Freedman and Waltman, 1984, Appendix 2). But the stable manifold of E, is the y axis which would be that orbit and is unbounded, a contradiction. If E, E Q, since it is a saddle point, it would have an orbit in its stable manifold belonging to Q. E,‘s stable manifold is the u-y plane, and all orbits in this plane are either unbounded or emanate from E,, giving a contradiction. Similarly, E2 x 51. If EE fi, the condition -s(G) + c(G) p(ii, 2) > 0 implies that i? is a saddle point. The stable manifold of ,!? is the U-X plane and hence orbits in this plane emanate from either E,, E,, or E, or are unbounded, once more a contradiction. Since no planar equilibrium is in the omega limit set, 8, no planar point is in Q since there are no other closed orbits in the planes. This proves the theorem.
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AND
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2
In order for E* to exist, the algebraic system Mu, 4 Y) = 0
UW
xg(u, x) - J&u, x) = 0
(15b)
-s(u) + c(u) p(u, x) = 0
(15c)
must have a positive solution. We first consider the curve given by (1%). From (4) and (5) this is a montonically decreasing curve in the u--x plane (see Fig. 3) and hence defines x as a function of U, x = q(u), such that either (I) there exists u0 > 0 such that lim, _ U; q(u) = +co (Fig. 3, case I), or (II) ~(0) =,? > 0 (Fig. 3, case II). In both cases dq/du
Y)
(18a)
r*: Y = f-P(u) A4 cp(U)YP(4 cp(u)).
(18b)
As far as ri is concerned, since dL/au = (aL/ax)(&p/au) 3 0, as u varies from zero to infinity, L(rp(u), u) for fixed y varies from L(m, y) to
CASE
MUTUALIST
II
(u)
FIG. 3. Case I: curve in the u-x plane defining q(u), determined by -s(u) where lim,,,+ q(u) = fco. Case II: curve in the u-x plane defining q(u), -s(u) + c(u) i(u, x) = 0, where lim,,,+ q(u) = P.
+ cp(u, x) = 0, determined by
STABILITY OF THREE-SPECIES MODELS
MUTUALIST
173
(u)
FIG. 4. r1 in case II when (a) aLjay < 0, (b) aL/@ = 0
L(q,, y) as u varies from q, to co in case I and L(& y) to L(cp,, y) in case II, where cpm= lim, _ m q(u). Hence, for fixed y, (18a) always has a solution u(y) in case II, but may or may not have such a solution in case I. If such a solution exists, since aLjay ~0, as y is increased, u is not increased for a solution (i.e., dy/du = (1 - aL/au)/(iTL/ay) < 0 if al;/@ < 0). We indicate f 1 in case II in Fig. 4, where on Z, , u = ri when y = 0. We now discuss Z2 restricting ourselves to case II. Utilizing (lk), (18b) may be written as
y = c(u) du) g(u, du)) s(u)
.
(19)
At u =O, y,,= c(0) 2g(O, 2)/s(O). y, > 0 if and only if 2
MUTUALIST
(u)
FIG. 5. Possible curves r,. Only in case (d) can E* exist.
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AND RAI
In case I, the analysis is even more complicated understanding of the problem.
and adds nothing to our
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