3. Models for Interacting Populations

Transcription

1 3. Models for Interacting Populations When species interact the population dynamics of each species is affected. In general there is a whole web of interacting species, sometimes called a trophic web, which makes for structurally complex communities. We consider here systems involving 2 or more species, concentrating particularly on two-species systems. The book by Kot (200) discusses such models (including age-structured interacting population systems) with numerous recent practical examples. There are three main types of interaction. (i) If the growth rate of one population is decreased and the other increased the populations are in a predator prey situation. (ii) If the growth rate of each population is decreased then it is competition. (iii) If each population s growth rate is enhanced then it is called mutualism or symbiosis. All of the mathematical techniques and analytical methods in this chapter are directly applicable to Chapter 6 on reaction kinetics, where similar equations arise; there the species are chemical concentrations. 3. Predator Prey Models: Lotka Volterra Systems Volterra (926) first proposed a simple model for the predation of one species by another to explain the oscillatory levels of certain fish catches in the Adriatic. If N(t) is the prey population and P(t) that of the predator at time t then Volterra s model is dn dp = N(a bp), (3.) = P(cN d), (3.2) where a, b, c and d are positive constants. The assumptions in the model are: (i) The prey in the absence of any predation grows unboundedly in a Malthusian way; this is the an term in (3.). (ii) The effect of the predation is to reduce the prey s per capita growth rate by a term proportional to the prey and predator populations; this is the bnp term. (iii) In the absence of any prey for sustenance the predator s death rate results in exponential decay, that is, the dp term in (3.2). (iv) The prey s contribution to the predators growth rate is cnp; that is, it is proportional to the available prey as well as to the size of the predator population. The NP terms can be thought of as representing the conversion of energy from one source

2 80 3. Models for Interacting Populations to another: bnp is taken from the prey and cnp accrues to the predators. We shall see that this model has serious drawbacks. Nevertheless it has been of considerable value in posing highly relevant questions and is a jumping-off place for more realistic models; this is the main motivation for studying it here. The model (3.) and (3.2) is known as the Lotka Volterra model since the same equations were also derived by Lotka (920; see also 925) from a hypothetical chemical reaction which he said could exhibit periodic behaviour in the chemical concentrations. With this motivation the dependent variables represent chemical concentrations; we touch on this again in Chapter 6. As a first step in analysing the Lotka Volterra model we nondimensionalise the system by writing and it becomes u(τ) = cn(t) d, v(τ) = bp(t), τ = at, α = d/a, (3.3) a du dτ = u( v), dv dτ = αv(u ). (3.4) In the u,v phase plane (a brief summary of basic phase plane methods is given in Appendix A) these give dv v(u ) = α du u( v), (3.5) which has singular points at u = v = 0andu = v =. We can integrate (3.5) exactly to get the phase trajectories αu + v ln u α v = H, (3.6) where H > H min is a constant: H min = + α is the minimum of H over all (u,v)and it occurs at u = v =. For a given H > + α, the trajectories (3.6) in the phase plane are closed as illustrated in Figure 3.. A closed trajectory in the u,v plane implies periodic solutions in τ for u and v in (3.4). The initial conditions, u(0) and v(0), determine the constant H in (3.6) and hence the phase trajectory in Figure 3.. Typical periodic solutions u(τ) and v(τ) are illustrated in Figure 3.2. From (3.4) we can see immediately that u has a turning point when v = andv has one when u =. A major inadequacy of the Lotka Volterra model is clear from Figure 3. the solutions are not structurally stable. Suppose, for example, u(0) and v(0) are such that u and v for τ>0 are on the trajectory H 4 which passes close to the u and v axes. Then any small perturbation will move the solution onto another trajectory which does not lie everywhere close to the original one H 4. Thus a small perturbation can have a very marked effect, at the very least on the amplitude of the oscillation. This is a problem with any system which has a first integral, like (3.6), which is a closed trajectory in the phase plane. They are called conservative systems; here (3.6) is the associated conservation

3 3. Predator Prey Models: Lotka Volterra Systems 8 v + α<h < H 2 < H 3 < H 4 H H 2 H 3 H 4 H 5 0 u Figure 3.. Closed (u,v)phase plane trajectories, from (3.6) with various H, for the Lotka Volterra system (3.4): H = 2., H 2 = 2.4, H 3 = 3.0, H 4 = 4. The arrows denote the direction of change with increasing time τ. law. They are usually of little use as models for real interacting populations (see one interesting and amusing attempt to do so below). However, the method of analysis of the steady states is typical. Returning to the form (3.4), a linearisation about the singular points determines the type of singularity and the stability of the steady states. A similar linear stability analysis has to be carried out on equivalent systems with any number of equations. We first consider the steady state (u,v) = (0, 0). Letx and y be small perturbations about (0, 0). If we keep only linear terms, (3.4) becomes The solution is of the form dx dτ dy dτ ( ) ( 0 x x = A. (3.7) 0 α)( y y) u (prey) v (predator) 0 τ Figure 3.2. Periodic solutions for the prey u(τ) and the predator v(τ) for the Lotka Volterra system (3.4) with α = and initial conditions u(0) =.25, v(0) = 0.66.

4 82 3. Models for Interacting Populations ( ) x(τ) = Be λτ, y(τ) where B is an arbitrary constant column vector and the eigenvalues λ are given by the characteristic polynomial of the matrix A and thus are solutions of A λi = λ 0 0 α λ = 0 λ =, λ 2 = α. Since at least one eigenvalue, λ > 0, x(τ) and y(τ) grow exponentially and so u = 0 = v is linearly unstable. Since λ > 0andλ 2 < 0 this is a saddle point singularity (see Appendix A). Linearising about the steady state u = v = by setting u = + x, v = + y with x and y small, (3.4) becomes with eigenvalues λ given by λ α dx dτ = A dy dτ ( x y), A = ( ) 0 α 0 (3.8) λ = 0 λ,λ 2 =±i α. (3.9) Thus u = v = isacentre singularity since the eigenvalues are purely imaginary. Since Re λ = 0 the steady state is neutrally stable. The solution of (3.8) is of the form ( ) x(τ) = le i ατ + me i ατ, y(τ) where l and m are eigenvectors. So, the solutions in the neighbourhood of the singular point u = v = are periodic in τ with period 2π/ α. In dimensional terms from (3.3) this period is T = 2π(a/d) /2 ; that is, the period is proportional to the square root of the ratio of the linear growth rate, a, of the prey to the death rate, d, of the predators. Even though we are only dealing with small perturbations about the steady state u = v = we see how the period depends on the intrinsic growth and death rates. For example, an increase in the growth rate of the prey will increase the period; a decrease in the predator death rate does the same thing. Is this what you would expect intuitively? In this ecological context the matrix A in the linear equations (3.7) and (3.8) is called the community matrix, and its eigenvalues λ determine the stability of the steady states. If Re λ>0 then the steady state is unstable while if both Re λ<0 it is stable. The critical case Re λ = 0istermedneutral stability. There have been many attempts to apply the Lotka Volterra model to real-world oscillatory phenomena. In view of the system s structural instability, they must essentially all fail to be of quantitative practical use. As we mentioned, however, they can be important as vehicles for suggesting relevant questions that should be asked. One particularly interesting example was the attempt to apply the model to the extensive data

5 3.2 Complexity and Stability 83 on the Canadian lynx snowshoe hare interaction in the fur catch records of the Hudson Bay Company from about 845 until the 930 s. We assume that the numbers reflect a fixed proportion of the total population of these animals. Although this assumption is of questionable accuracy, as indicated by what follows, the data nevertheless represent one of the very few long term records available. Figure 3.3 reproduces this data. Williamson s (996) book is a good source of population data which exhibit periodic or quasi-periodic behaviour. Figure 3.3 shows reasonable periodic fluctuations and Figure 3.3(c) a more or less closed curve in the phase plane as we now expect from a time-periodic behaviour in the variables. Leigh (968) used the standard Lotka Volterra model to try to explain the data. Gilpin (973) did the same with a modified Lotka Volterra system. Let us examine the results given in Figure 3.3 a little more carefully. First note that the direction of the time arrows in Figure 3.3(c) is clockwise in contrast to that in Figure 3.. This is reflected in the time curves in Figures 3.3(a) and (b) where the lynx oscillation, the predator s, precedes the hare s. The opposite is the case in the predator prey situation illustrated in Figure 3.2. Figure 3.3 implies that the hares are eating the lynx! This poses a severe interpretation problem! Gilpin (973) suggested that perhaps the hares could kill the lynx if they carried a disease which they passed on to the lynx. He incorporated an epidemic effect into his model and the numerical results then looked like those in Figure 3.3(c); this seemed to provide the explanation for the hare eating the lynx. A good try, but no such disease is known. Gilpin (973) also offered what is perhaps the right explanation, namely, that the fur trappers are the disease. In years of low population densities they probably did something else and only felt it worthwhile to return to the trap lines when the hares were again sufficiently numerous. Since lynx were more profitable to trap than hare they would probably have devoted more time to the lynx than the hare. This would result in the phenomenon illustrated by Figures 3.3(b) and (c). Schaffer (984) has suggested that the lynx hare data could be evidence of a strange attractor (that is, they exhibit chaotic behaviour) in Nature. The moral of the story is that it is not enough simply to produce a model which exhibits oscillations but rather to provide a proper explanation of the phenomenon which can stand up to ecological and biological scrutiny. 3.2 Complexity and Stability To get some indication of the effect of complexity on stability we briefly consider the generalised Lotka Volterra predator prey system where there are k prey species and k predators, which prey on all the prey species but with different severity. Then in place of (3.) and (3.2) we have ] dn i k = N i [a i b ij P j j= i =,...,k (3.0) dp i [ ] k = P i c ij N j d i, j=

6 84 3. Models for Interacting Populations Figure 3.3. (a) Fluctuations in the number of pelts sold by the Hudson Bay Company. (Redrawn from Odum 953) (b) Detail of the 30-year period starting in 875, based on the data from Elton and Nicholson (942). (c) Phase plane plot of the data represented in (b). (After Gilpin 973)

7 3.2 Complexity and Stability 85 where all of the a i, b ij, c ij and d i are positive constants. The trivial steady state is N i = P i = 0foralli, and the community matrix is the diagonal matrix The 2k eigenvalues are thus a a k A =. d d k λ i = a i > 0, λ k+i = d i < 0, i =,...,k so this steady state is unstable since all λ i > 0, i =,...,k. The nontrivial steady state is the column vector solution N, P where k b ij Pj = a i, j= k c ij N j = d i, i =,...,k j= or, in vector notation, with N, P, a, andd column vectors, BP = a, CN = d, (3.) where B and C are the k k matrices [b ij ] and [c ij ] respectively. Equations (3.0) can be written as dn = N T [a BP], dp = P T [CN d], where the superscript T denotes the transpose. So, on linearising about (N, P ) in (3.) by setting N = N + u, P = P + v, where u, v are small compared with N and P,weget Then du N T Bv, dv P T Cu. du ( ( u 0 N T ) B A, A =, (3.2) dv v) P T C 0

8 86 3. Models for Interacting Populations where here the community matrix A is a 2k 2k block matrix with null diagonal blocks. Since the eigenvalues λ i, i =,...,2k are solutions of A λi =0 the sum of the roots λ i satisfies 2k i= λ i = tra = 0, (3.3) where tra is the trace of A. Since the elements of A are real, the eigenvalues, if complex, occur as complex conjugates. Thus from (3.3) there are two cases: all the eigenvalues are purely imaginary or they are not. If all Re λ i = 0 then the steady state (N, P ) is neutrally stable as in the 2-species case. However if there are λ i such that Re λ i = 0 then, since they occur as complex conjugates, (3.3) implies that at least one exists with Re λ>0 and hence (N, P ) is unstable. We see from this analysis that complexity in the population interaction web introduces the possibility of instability. If a model by chance resulted in only imaginary eigenvalues (and hence perturbations from the steady state are periodic in time) only a small change in one of the parameters in the community matrix would result in at least one eigenvalue with Re λ = 0 and hence an unstable steady state. This of course only holds for community matrices such as in (3.2). Even so, we get indications of the fairly general and important result that complexity usually results in instability rather than stability. 3.3 Realistic Predator Prey Models The Lotka Volterra model, unrealistic though it is, does suggest that simple predator prey interactions can result in periodic behaviour of the populations. Reasoning heuristically this is not unexpected since if a prey population increases, it encourages growth of its predator. More predators however consume more prey the population of which starts to decline. With less food around the predator population declines and when it is low enough, this allows the prey population to increase and the whole cycle starts over again. Depending on the detailed system such oscillations can grow or decay or go into a stable limit cycle oscillation or even exhibit chaotic behaviour, although in the latter case there must be at least 3 interacting species, or the model has to have some delay terms. A limit cycle solution is a closed trajectory in the predator prey space which is not a member of a continuous family of closed trajectories such as the solutions of the Lotka Volterra model illustrated in Figure 3.. A stable limit cycle trajectory is such that any small perturbation from the trajectory decays to zero. A schematic example of a limit cycle trajectory in a two-species predator(p) prey(n) interaction is illustrated in Figure 3.4. Conditions for the existence of such a solution are given in Appendix A. One of the unrealistic assumptions in the Lotka Volterra models, (3.) and (3.2), and generally (3.0), is that the prey growth is unbounded in the absence of predation. In the form we have written the model (3.) and (3.2) the bracketed terms on the right are the density-dependent per capita growth rates. To be more realistic these growth

9 3.3 Realistic Predator Prey Models 87 Figure 3.4. Typical closed predator prey trajectory which implies a limit cycle periodic oscillation. Any perturbation from the limit cycle tends to zero asymptotically with time. rates should depend on both the prey and predator densities as in dn = NF(N, P), dp = PG(N, P), (3.4) where the forms of F and G depend on the interaction, the species and so on. As a reasonable first step we might expect the prey to satisfy a logistic growth, say, in the absence of any predators, that is, like (.2) in Chapter, or have some similar growth dynamics which has some maximum carrying capacity. So, for example, a more realistic prey population equation might take the form dn = NF(N, P), F(N, P) = r ( N ) PR(N), (3.5) K where R(N) is one of the predation terms discussed below and illustrated in Figure 3.5 and K is the constant carrying capacity for the prey when P 0. The predation term, which is the functional response of the predator to change in the prey density, generally shows some saturation effect. Instead of a predator response of bnp, as in the Lotka Volterra model (3.), we take PNR(N) where NR(N) saturates for N large. Some examples are Figure 3.5. Examples of predator response NR(N) to prey density N.(a) R(N) = A, the unsaturated Lotka Volterra type. (b) R(N) = A/(N + B).(c) R(N) = AN/(N 2 + B 2 ).(d) R(N) = A( e an )/N.

10 88 3. Models for Interacting Populations R(N) = A AN, R(N) = N + B N 2 + B 2, R(N) = A[ e an ], (3.6) N where A, B and a are positive constants; these are illustrated in Figures 3.5(b) to (d). The second of (3.6), illustrated in Figure 3.5(c), is similar to that used in the budworm model in equation (.6) in Chapter. It is also typical of aphid (Aphidicus zbeckistanicus) predation. The examples in Figures 3.5(b) and (c) are approximately linear in N for low densities. The saturation for large N is a reflection of the limited predator capability, or perseverance, when the prey is abundant. The predator population equation, the second of (3.4), should also be made more realistic than simply having G = d + cn as in the Lotka Volterra model (3.2). Possible forms are G(N, P) = k ( hp N ), G(N, P) = d + er(n), (3.7) where k, h, d and e are positive constants and R(N) is as in (3.6). The first of (3.7) says that the carrying capacity for the predator is directly proportional to the prey density. The models given by (3.4) (3.7) are only examples of the many that have been proposed and studied. They are all more realistic than the classical Lotka Volterra model. Other examples are discussed, for example, in the book by Nisbet and Gurney (982) and that edited by Levin (994), to mention but two. 3.4 Analysis of a Predator Prey Model with Limit Cycle Periodic Behaviour: Parameter Domains of Stability As an example of how we analyze such realistic 2-species models we consider one of them in detail, namely, [ ( dn = N r N ) kp ], K N + D [ ( dp = P s hp )] (3.8), N where r, K, k, D, s and h are positive constants, 6 in all. It is, as always, extremely useful to write the system in nondimensional form. Although there is no unique way of doing this it is often a good idea to relate the variables to some key relevant parameter. Here, for example, we express N and P as fractions of the predator-free carrying capacity K. Let us write and (3.8) become u(τ) = N(t) hp(t), v(τ) = K K, τ = rt, a = k hr, b = s r, d = D K (3.9)

11 3.4 Analysis of a Predator Prey Model with Limit Cycle Periodic Behaviour 89 du auv = u ( u) dτ u + d = f (u,v), dv ( dτ = bv v ) = g(u,v), u (3.20) which have only 3 dimensionless parameters a, b and d. Nondimensionalisation reduces the number of parameters by grouping them in a meaningful way. Dimensionless groupings generally give relative measures of the effect of dimensional parameters. For example, b is the ratio of the linear growth rate of the predator to that of the prey and so b > andb < have definite ecological meanings; with the latter the prey reproduce faster than the predator. The equilibrium or steady state populations u,v are solutions of du/dτ = 0, dv/dτ = 0; namely, f (u,v ) = 0, g(u,v ) = 0 which, from the last equations, are u ( u ) au v u + d = 0, ) ( bv v = 0. (3.2) u We are only concerned here with positive solutions, namely, the positive solutions of of which the only positive one is v = u, u 2 + (a + d )u d = 0, u = ( a d) +{( a d)2 + 4d} /2, v = u. (3.22) 2 We are interested in the stability of the steady states, which are the singular points in the phase plane of (3.20). A linear stability analysis about the steady states is equivalent to the phase plane analysis. For the linear analysis write x(τ) = u(τ) u, y(τ) = v(τ) v (3.23) which on substituting into (3.20), linearising with x and y small, and using (3.2), gives f A = u g u dx dτ = A dy dτ f v g = v u,v u ( x y), au ] (u + d) 2 [ b au u + d. b (3.24)

12 90 3. Models for Interacting Populations A, the community matrix, has eigenvalues λ given by A λi =0 λ 2 (tra)λ + det A = 0. (3.25) For stability we require Re λ<0and so the necessary and sufficient conditions for linear stability are, from the last equation, [ tra < 0 u au ] (u + d) 2 < b, det A > 0 + a (3.26) u + d au (u + d) 2 > 0. Substituting for u from (3.22) gives the stability conditions in terms of the parameters a, b and d, and hence in terms of the original parameters r, K, k, D, s and h in (3.8). In general there is a domain in the a, b, d space such that, if the parameters lie within it, (u,v ) is stable, that is, Re λ<0, and if they lie outside it the steady state is unstable. The latter requires at least one of (3.26) to be violated. With (3.22) for u and using the first of (3.2) and v = u, [ det A = + a u + d au (u + d) [ ] 2 = + bu > 0 ad (u + d) 2 ] bu (3.27) for all a > 0, b > 0, d > 0 and so the second of (3.26) is always satisfied. The instability domain is thus determined solely by the first inequality of (3.26), namely, tr A < 0 which, with (3.22) for u and again using (3.2), becomes [ [ b > a {( a d) 2 + 4d} /2] + a + d {( a d) 2 + 4d} /2]. (3.28) 2a This defines a three-dimensional surface in (a, b, d) parameter space. We are only concerned with a, b, and d positive. The second square bracket in (3.28) is a monotonic decreasing function of d and always positive. The first square bracket is a monotonic decreasing function of d with a maximum at d = 0. Thus, from (3.28), b d=0 { > 2a > /a if { 0 < a a and so for 0 < a < /2andalld > 0 the stability condition (3.28) is satisfied with any b > 0. That is, the steady state u,v is linearly stable for all 0 < a < /2, b > 0, d > 0. On the other hand if a > /2 there is a domain in the (a, b, d) space with b > 0 and d > 0 where (3.28) is not satisfied and so the first of (3.26) is violated and hence

13 3.4 Analysis of a Predator Prey Model with Limit Cycle Periodic Behaviour 9 one of the eigenvalues λ in (3.25) has Re λ>0. This in turn implies the steady state u,v is unstable to small perturbations. The boundary surface is given by (3.28) and it crosses the b = 0 plane at d = d m (a) given by the positive solution of a ={( a d m )2 + 4d m } /2 d m (a) = d b=0 = (a 2 + 4a) /2 ( + a). Thus d m (a) is a monotonic increasing function of a bounded above by d =. Note also that d < a for all a > /2. Figure 3.6 illustrates the stability/instability domains in the (a, b, d) space. When Re λ<0 the steady state is stable and either both λ s are real in (3.25), in which case the singular point u,v in (3.2) is a stable node in the u,v phase plane of (3.20), or the λ s are complex and the singular point is a stable spiral. When the parameters result in Re λ>0 the singular point is either an unstable node or spiral. In this case we must determine whether or not there is a confined set, or bounding domain, in the (u,v)phase plane so as to use the Poincaré Bendixson theorem for the existence of a limit cycle oscillation; see Appendix A. In other words we must find a simple closed boundary curve in the positive quadrant of the (u,v) plane such that on it the phase trajectories always point into the enclosed domain. That is, if n denotes the outward normal to this boundary, we require Figure 3.6. Parameter domains (schematic) of stability of the positive steady state for the predator prey model (3.20). For a < /2 and all parameter values b > 0, d > 0, stability is obtained. For a fixed a > /2, the domain of instability is finite as in (a)and(b). The three-dimensional bifurcation surface between stability and instability is sketched in (c) with d m (a) = (a 2 + 4a) /2 ( + a). When parameter values are in the unstable domain, limit cycle oscillations occur in the populations.

14 92 3. Models for Interacting Populations ( du n dτ, dv ) < 0 dτ for all points on the boundary. If this inequality holds at a point on the boundary it means that the velocity vector (du/dτ,dv/dτ) points inwards. Intuitively this means that no solution trajectory can leave the domain if once inside, since, if it did reach the boundary, its velocity points inwards and so the trajectory moves back into the domain. To find a confined set it is essential and always informative to draw the null clines of the system, that is, the curves in the phase plane where du/dτ = 0anddv/dτ = 0. From (3.20) these are the curves f (u,v) = 0andg(u,v) = 0 which are illustrated in Figure 3.7. The sign of the vector components of ( f (u,v),g(u,v))indicate the direction of the vector (du/dτ,dv/dτ) and hence the direction of the (u,v)trajectory. So if f > 0 in a domain, du/dτ >0andu is thus increasing there. On DE, EA, AB and BC, the trajectories clearly point inwards because of the signs of f (u,v) and g(u,v) on them. It can be shown simply but tediously that a line DC exists such that on it n (du/dτ,dv/dτ) < 0; that is, n ( f (u,v),g(u,v)) < 0 where n is the unit vector perpendicular to DC. We now have a confined set appropriate for the Poincaré Bendixson theorem to apply when (u,v ) is unstable. Hence the solution trajectory tends to a limit cycle when the parameters a, b and d lie in the unstable domain in Figure 3.6(c). Basically the Poincaré Bendixson theorem says that since any trajectory coming out of the unstable steady state (u,v ) cannot cross the confining boundary ABCDEA,itmust evolve into a closed limit cycle trajectory qualitatively similar to that illustrated in Figure 3.4. With our model (3.20), Figure 3.8(a) illustrates such a closed trajectory with Figure 3.8(b) showing the temporal variation of the populations with time. With the specific parameter values used in Figure 3.8 the steady state is an unstable node in the phase plane; that is, both eigenvalues are real and positive. Any perturbation from the limit cycle decays quickly. Figure 3.7. Null clines f (u,v) = 0, g(u,v) = 0 for the system (3.20); note the signs of f and g on either side of their null clines. ABCDEA is the boundary of the confined set about (u,v ) on which the trajectories all point inwards; that is, n (du/dτ, dv/dτ) < 0wheren is the unit outward normal on the boundary ABCDEA.

15 3.4 Analysis of a Predator Prey Model with Limit Cycle Periodic Behaviour 93 Figure 3.8. (a) Typical phase trajectory limit cycle solution for the predator prey system (3.20). (b) Corresponding periodic behaviour of the prey (u) and predator (v) populations. Parameter values: a =, b = 5, d = 0.2, which give the steady state as u = v = Relations (3.9) relate the dimensionless to the dimensional parameters. This model system, like most which admit limit cycle behaviour, exhibits bifurcation properties as the parameters vary, although not with the complexity shown by discrete models as we see in Chapters 2 and 5, nor with delay models such as in Chapter. We can see this immediately from Figure 3.6. To be specific, consider a fixed a > /2 so that a finite domain of instability exists, as illustrated in Figure 3.9, and let us choose a fixed 0 < d < d m corresponding to the line DEF. Suppose b is initially at the value D and is then continuously decreased. On crossing the bifurcation line at E, the steady state becomes unstable and a periodic limit cycle solution appears; that is, the uniform steady state bifurcates to an oscillatory solution. A similar situation occurs along any parameter variation from the stable to the unstable domains in Figure 3.6(c). The fact that a dimensionless variable passes through a bifurcation value provides useful practical information on equivalent effects of dimensional parameters. For example, from (3.9), b = s/r, the ratio of the linear growth rates of the predator and prey. If the steady state is stable, then as the predators growth rate s decreases there is more likelihood of periodic behaviour since b decreases and, if it decreases enough, we move into the instability regime. On the other hand if r decreases, b increases and so probably reduces the possibility of oscillatory behaviour. In this latter case it is not so clear-cut since, from (3.9), reducing r also increases a, which from Figure 3.6(c) tends to increase the possibility of periodic behaviour. The dimensional bifurcation space is 6-dimensional which is difficult to express graphically; the nondimensionalisation reduces it to a simple 3-dimensional space with (3.9) giving clear equivalent effects of Figure 3.9. Typical stability bifurcation curve for the predator prey model (3.20). As the point in parameter space crosses the bifurcation curve, the steady state changes stability.

16 94 3. Models for Interacting Populations different dimensional parameter changes. For example, doubling the carrying capacity K is exactly equivalent to halving the predator response parameter D. The dimensionless parameters are the important bifurcation ones to determine. 3.5 Competition Models: Principle of Competitive Exclusion Here two or more species compete for the same limited food source or in some way inhibit each other s growth. For example, competition may be for territory which is directly related to food resources. Some interesting phenomena have been found from the study of practical competition models; see, for example, Hsu et al. (979). Here we discuss a very simple competition model which demonstrates a fairly general principle which is observed to hold in Nature, namely, that when two species compete for the same limited resources one of the species usually becomes extinct. Consider the basic 2-species Lotka Volterra competition model with each species N and N 2 having logistic growth in the absence of the other. Inclusion of logistic growth in the Lotka Volterra systems makes them much more realistic, but to highlight the principle we consider the simpler model which nevertheless reflects many of the properties of more complicated models, particularly as regards stability. We thus consider [ dn = r N N ] N 2 b 2, (3.29) K dn 2 K = r 2 N 2 [ N 2 K 2 b 2 N K 2 ], (3.30) where r, K, r 2, K 2, b 2 and b 2 are all positive constants and, as before, the r s are the linear birth rates and the K s are the carrying capacities. The b 2 and b 2 measure the competitive effect of N 2 on N and N on N 2 respectively: they are generally not equal. Note that the competition model (3.29) and (3.30) is not a conservative system like its Lotka Volterra predator prey counterpart. If we nondimensionalise this model by writing u = N K, u 2 = N 2 K 2, τ = r t, ρ = r 2 r, a 2 = b 2 K 2 K, a 2 = b 2 K K 2 (3.3) (3.29) and (3.30) become du dτ = u ( u a 2 u 2 ) = f (u, u 2 ), du 2 dτ = ρu 2( u 2 a 2 u ) = f 2 (u, u 2 ). (3.32) The steady states, and phase plane singularities, u, u 2, are solutions of f (u, u 2 ) = f 2 (u, u 2 ) = 0 which, from (3.32), are

17 3.5 Competition Models: Competitive Exclusion Principle 95 u = 0, u 2 = 0; u =, u 2 = 0; u = 0, u 2 = ; u = a 2, u 2 a 2 a = a 2. 2 a 2 a 2 (3.33) The last of these is only of relevance if u 0andu 2 0 are finite, in which case a 2 a 2 =. The four possibilities are seen immediately on drawing the null clines f = 0and f 2 = 0intheu, u 2 phase plane as shown in Figure 3.0. The crucial part of the null clines are, from (3.32), the straight lines u a 2 u 2 = 0, u 2 a 2 u = 0. The first of these together with the u 2 -axis is f = 0, while the second, together with the u -axis is f 2 = 0. The stability of the steady states is again determined by the community matrix which, for (3.32), is u 2 (a) u 2 (b) a 2 f <0 u 2 a 2 u =0 f >0 u a 2 u 2 =0 a 2 f 2 <0 0 f 2 >0 0 u a 2 a 2 u u 2 a 2 (c) u 2 (d) a 2 0 a 2 u 0 u a 2 Figure 3.0. The null clines for the competition model (3.32). f = 0isu = 0and u a 2 u 2 = 0 with f 2 = 0beingu 2 = 0and u 2 a 2 u = 0. The intersection of the two solid lines gives the positive steady state if it exists as in (a)and(b): the relative sizes of a 2 and a 2 as compared with for it to exist are obvious from (a) to(d).

18 96 3. Models for Interacting Populations f f A = u u 2 f 2 f 2 u u 2 u,u 2 ( ) 2u a = 2 u 2 a 2 u ρa 2 u 2 ρ( 2u 2 a 2 u ) u,u 2. (3.34) The first steady state in (3.33), that is, (0, 0), is unstable since the eigenvalues λ of its community matrix, given from (3.34) by A λi = λ 0 0 ρ λ = 0 λ =,λ 2 = ρ, are positive. For the second of (3.33), namely, (, 0), (3.34) gives and so A λi = λ a 2 0 ρ( a 2 ) λ = 0 λ =, λ 2 = ρ( a 2 ) u =, u 2 = 0 is { stable unstable if { a2 > a 2 <. (3.35) Similarly, for the third steady state, (0, ), the eigenvalues are λ = ρ,λ 2 = ( a 2 ) and so u = 0, u 2 = is { stable unstable if { a2 > a 2 <. (3.36) Finally for the last steady state in (3.33), when it exists in the positive quadrant, the matrix A from (3.34) is which has eigenvalues ( ) A = ( a 2 a 2 ) a2 a 2 (a 2 ) ρa 2 (a 2 ) ρ(a 2 ) λ,λ 2 = [ 2( a 2 a 2 ) ] [ (a 2 ) + ρ(a 2 ) ± {[ (a 2 ) + ρ(a 2 ) ] 2 4ρ( a2 a 2 )(a 2 )(a 2 ) } /2]. (3.37) The sign of λ, orreλ if complex, and hence the stability of the steady state, depends on the size of ρ,a 2 and a 2. There are several cases we have to consider, all of which have ecological implications which we come to below.

19 3.5 Competition Models: Competitive Exclusion Principle 97 Before discussing the various cases note that there is a confined set on the boundary of which the vector of the derivatives, (du /dτ,du 2 /dτ), points along it or inwards: here it is a rectangular box in the (u, u 2 ) plane. From (3.32) this condition holds on the u -andu 2 -axes. Outer edges of the rectangle are, for example, the lines u = U where U a 2 u 2 < 0 and u 2 = U 2 where U 2 a 2 u < 0. Any U >, U 2 > suffice. So the system is always globally stable. The various cases are: (i) a 2 <, a 2 <, (ii) a 2 >, a 2 >, (iii) a 2 <, a 2 >, (iv) a 2 >, a 2 <. All of these are analyzed in a similar way. Figures 3.0(a) to (d) and Figures 3.(a) to (d) relate to these cases (i) to (iv) respectively. By way of example, we consider just one of them, namely, (ii). The analysis of the other cases is left as an exercise. The results are encapsulated in Figure 3.. The arrows in- u 2 (a) u 2 (b) a 2 II Separatrix u 2 S a 2 u 2 I 0 u 0 u u a 2 a2 u u 2 (c) u 2 (d) a 2 a 2 0 u 0 a 2 Figure 3.. Schematic phase trajectories near the steady states for the dynamic behaviour of competing populations satisfying the model (3.32) for the various cases. (a) a 2 <, a 2 <. Only the positive steady state S is stable and all trajectories tend to it. (b) a 2 >, a 2 >. Here, (, 0) and (0, ) are stable steady states, each of which has a domain of attraction separated by a separatrix which passes through (u, u 2 ).(c) a 2 <, a 2 >. Only one stable steady state exists, u =, u 2 = 0 with the whole positive quadrant its domain of attraction. (d) a 2 >, a 2 <. The only stable steady state is u = 0, u 2 = with the positive quadrant as its domain of attraction. Cases (b) to(d) illustrate the competitive exclusion principle whereby 2 species competing for the same limited resource cannot in general coexist. a2 u

20 98 3. Models for Interacting Populations dicate the direction of the phase trajectories. The qualitative behaviour of the phase trajectories is given by the signs of du /dτ, namely, f (u, u 2 ),anddu 2 /dτ which is f 2 (u, u 2 ), on either side of the null clines. Case a 2 >, a 2 >. This corresponds to Figure 3.0(b). From (3.35) and (3.36), (, 0) and (0, ) are stable. Since a 2 a 2 < 0, (u, u 2 ), the fourth steady state in (3.33), lies in the positive quadrant and from (3.37) its eigenvalues are such that λ 2 < 0 <λ and so it is unstable to small perturbations: it is a saddle point. In this case, then, the phase trajectories can tend to either one of the two steady states, as illustrated in Figure 3.(b). Each steady state has a domain of attraction. There is a line, a separatrix, which divides the positive quadrant into 2 nonoverlapping regions I and II as in Figure 3.(b). The separatrix passes through the steady state (u, u 2 ):itis one of the saddle point trajectories in fact. Now consider some of the ecological implications of these results. In case (i) where a 2 < anda 2 < there is a stable steady state where both species can exist as in Figure 3.0(a). In terms of the original parameters from (3.3) this corresponds to b 2 K 2 /K < and b 2 K /K 2 <. For example, if K and K 2 are approximately the same and the interspecific competition, as measured by b 2 and b 2, is not too strong, these conditions say that the two species simply adjust to a lower population size than if there were no competition. In other words, the competition is not aggressive. On the other hand if the b 2 and b 2 are about the same and the K and K 2 are different, it is not easy to tell what will happen until we form and compare the dimensionless groupings a 2 and a 2. In case (ii), where a 2 > anda 2 >, if the K s are about equal, then the b 2 and b 2 are not small. The analysis then says that the competition is such that all three nontrivial steady states can exist, but, from (3.35) to (3.37), only (, 0) and (0, ) are stable, as in Figure 3.(b). It can be a delicate matter which ultimately wins out. It depends crucially on the starting advantage each species has. If the initial conditions lie in domain I then eventually species 2 will die out, u 2 0andu ; that is, N K the carrying capacity of the environment for N. Thus competition here has eliminated N 2. On the other hand if N 2 has an initial size advantage so that u and u 2 start in region II then u 0 and u 2 in which case the N -species becomes extinct and N 2 K 2, its environmental carrying capacity. We expect extinction of one species even if the initial populations are close to the separatrix and in fact if they lie on it, since the ever present random fluctuations will inevitably cause one of u i, i =, 2toteno zero. Cases (iii) and (iv) in which the interspecific competition of one species is much stronger than the other, or the carrying capacities are sufficiently different so that a 2 = b 2 K 2 /K < anda 2 = b 2 K /K 2 > or alternatively a 2 > anda 2 <, are quite definite in the ultimate result. In case (iii), as in Figure 3.(c), the stronger dimensionless interspecific competition of the u -species dominates and the other species, u 2, dies out. In case (iv) it is the other way round and species u becomes extinct. Although all cases do not result in species elimination, those in (iii) and (iv) always do and in (ii) it is inevitable due to natural fluctuations in the population levels. This work led to the principle of competitive exclusion which was mentioned above. Note that the conditions for this to hold depend on the dimensionless parameter groupings a 2 and a 2 : the growth rate ratio parameter ρ does not affect the gross stability results, just

21 3.6 Mutualism or Symbiosis 99 the dynamics of the system. Since a 2 = b 2 K 2 /K, a 2 = b 2 K /K 2 the conditions for competitive exclusion depend critically on the interplay between competition and the carrying capacities as well as the initial conditions in case (ii). Suppose, for example, we have 2 species comprised of large animals and small animals, with both competing for the same grass in a fixed area. Suppose also that they are equally competitive with b 2 = b 2. With N the large animals and N 2 the small, K < K 2 and so a 2 = b 2 K 2 /K < b 2 K 2 /K = a 2. As an example if b 2 = = b 2, a 2 < anda 2 > then in this case N 0andN 2 K 2 ; that is, the large animals become extinct. The situation in which a 2 = = a 2 is special and, with the usual stochastic variability in nature, is unlikely in the real world to hold exactly. In this case the competitive exclusion of one or the other of the species also occurs. The importance of species competition in Nature is obvious. We have discussed only one particularly simple model but again the method of analysis is quite general. A review and introductory article by Pianka (98) deals with some practical aspects of competition as does the book of lectures by Waltman (984). A slightly simpler competition model (see Exercise 2) was applied by Flores (998) to the extinction of Neanderthal man by Early Modern man. Flores model is based on a slightly different mortality rate of the two species and he shows that coexistence is not possible. He estimates the relevant parameter from independent sources and his extinction period is in line with the accepted palaeontological data of 5000 to 0,000 years. In Chapters and 4, Volume II we discuss some practical cases of spatial competition associated with squirrels, wolf deer survival and the release of genetically engineered organisms. 3.6 Mutualism or Symbiosis There are many examples where the interaction of two or more species is to the advantage of all. Mutualism or symbiosis often plays the crucial role in promoting and even maintaining such species: plant and seed dispersal is one example. Even if survival is not at stake the mutual advantage of mutualism or symbiosis can be very important. As a topic of theoretical ecology, even for two species, this area has not been as widely studied as the others even though its importance is comparable to that of predator prey and competition interactions. This is in part due to the fact that simple models in the Lotka Volterra vein give silly results. The simplest mutualism model equivalent to the classical Lotka Volterra predator prey one is dn = r N + a N N 2, dn 2 = r 2 N 2 + a 2 N 2 N, where r, r 2, a and a 2 are all positive constants. Since dn / > 0anddN 2 / > 0, N and N 2 simply grow unboundedly in, as May (98) so aptly put it, an orgy of mutual benefaction. Realistic models must at least show a mutual benefit to both species, or as many as are involved, and have some positive steady state or limit cycle type oscillation.

22 00 3. Models for Interacting Populations Some models which do this are described by Whittaker (975). A practical example is discussed by May (975). As a first step in producing a reasonable 2-species model we incorporate limited carrying capacities for both species and consider ( dn = r N N ) N 2 + b 2 K dn 2 K = r 2 N 2 ( N 2 K 2 + b 2 N K 2 ), (3.38) where r, r 2, K, K 2, b 2 and b 2 are all positive constants. If we use the same nondimensionalisation as in the competition model (the signs preceding the b s are negative there), namely, (3.3), we get where du dτ = u ( u a 2 u 2 ) = f (u, u 2 ), du 2 dτ = ρu 2( u 2 a 2 u ) = f 2 (u, u 2 ), (3.39) u = N K, u 2 = N 2 K 2, τ = r t, ρ = r 2 r, a 2 = b 2 K 2 K, a 2 = b 2 K K 2. (3.40) Analysing the model in the usual way we start with the steady states (u, u 2 ) which from (3.39) are ( + a2 δ (0, 0), (, 0), (0, ),, + a ) 2, positive if δ = a 2 a 2 > 0. δ (3.4) After calculating the community matrix for (3.39) and evaluating the eigenvalues λ for each of (3.4) it is straightforward to show that (0, 0), (, 0) and (0, ) are all unstable: (0, 0) is an unstable node and (, 0) and (0, ) are saddle point equilibria. If a 2 a 2 < 0 there are only three steady states, the first three in (3.4), and so the populations become unbounded. We see this by drawing the null clines in the phase plane for (3.39), namely, f = 0, f 2 = 0, and noting that the phase trajectories move off to infinity in a domain in which u and u 2 as in Figure 3.2(a). When a 2 a 2 > 0, the fourth steady state in (3.4) exists in the positive quadrant. Evaluation of the eigenvalues of the community matrix shows it to be a stable equilibrium: it is a node singularity in the phase plane. This case is illustrated in Figure 3.2(b). Here all the trajectories in the positive quadrant tend to u > andu 2 > ; that is, N > K and N 2 > K 2 and so each species has increased its steady state population from its maximum value in isolation. This model has certain drawbacks. One is the sensitivity between unbounded growth and a finite positive steady state. It depends on the inequality a 2 a 2 <, which from

23 3.7 General Models and Cautionary Remarks 0 u 2 f 2 < 0 u 2 + a 2 u = 0 u 2 f 2 > 0 u 2 S u + a 2 u 2 = 0 f > 0 f < 0 0 u 0 u (a) Figure 3.2. Phase trajectories for the mutualism model for two species with limited carrying capacities given by the dimensionless system (3.39). (a) a 2 a 2 > : unbounded growth occurs with u and u 2 in the domain bounded by the null clines the solid lines. (b) a 2 a 2 < : all trajectories tend to a positive steady state S with u >, u 2 > which shows the initial benefit that accrues since the carrying capacity for each species is greater than if no interaction were present. (b) u (3.40) in terms of the original parameters in (3.38) is b 2 b 2 < : the b s are dimensionless. So if symbiosis of either species is too large, this last condition is violated and both populations grow unboundedly. 3.7 General Models and Some General and Cautionary Remarks All of the models we have discussed in this chapter result in systems of nonlinear differential equations of the form dn i = N i F i (N, N 2,...,N n ), i =, 2,..., (3.42) which emphasises the fact that the vector of populations N has N = 0 as a steady state. The two-species version is sometimes referred to as the Kolmogorov model or as the Kolmogorov equations. Although we have mainly considered 2-species interactions in this chapter, in nature, and in the sea in particular, there are many species or trophic levels where energy, in the form of food, flows from one species to another. That is, there is a flow from one trophic level to another. The mass of the total number of individuals in a species is often referred to as its biomass, here the population times the unit mass. The ultimate source of energy is the sun, and in the sea, for example, the trophic web runs through plankton, fish, sharks up to whales and finally man, with the myriad of species in between. The species on one trophic level may predate several species below it. In general, models involve interaction between several species.

24 02 3. Models for Interacting Populations Multi-species models are of the form du = f(u) or du i = f i (u,...,u n ), i =,...,n, (3.43) where u(t) is the n-dimensional vector of population densities and f(u) describes the nonlinear interaction between the species. The function f(u) involves parameters which characterise the various growth and interaction features of the system under investigation, with f i (u,...,u n ) specifying the overall rate of growth for the ith species. The stability of the steady states is determined in exactly the same way as before by linearising about the steady states u, where f(u ) = 0 and examining the eigenvalues λ of the community or stability matrix ( ) fi A = (a ij ) =. (3.44) u j u=u The necessary and sufficient conditions for the eigenvalues λ, solutions of polynomial A λi =0, to have Re λ>0are given by the Routh Hurwitz conditions which are listed in Appendix B. If a steady state is unstable then the solution u may grow unboundedly or evolve into another steady state or into a stable oscillatory pattern like a limit cycle. For 2- species models the theory of such equations is essentially complete: they are phase plane systems and a brief review of their analysis is given in Appendix A. For three or more interacting species considerably less general theory exists. Some results, at least for solutions near the steady state when it becomes unstable, can often be found using Hopf bifurcation theory; see, for example, the book by Strogatz (994) for a good pedagogical discussion of Hopf bifurcation. At its simplest this theory says that if a parameter of the system, p say, has a critical value p c such that for p < p c the eigenvalue with the largest Re λ<0, and for p = p c Re λ = 0, Im λ = 0andforp > p c Re λ>0, Im λ = 0 then for p p c > 0 and small, the solution u will exhibit small amplitude limit cycle behaviour around u. Smith (993) has developed a new approach to the study of 3 (and higher) competitive or cooperative species. His approach lets you apply the Poincaré Bendixson theorem to three-species systems by relating the flows to topologically equivalent flows in two dimensions. The community matrix A, defined by (3.44), which is so crucial in determining the linear stability of the steady states, has direct biological significance. The elements a ij measure the effect of the j-species on the i-species near equilibrium. For example, if U i is the perturbation from the steady state ui, the equation for U i is du i = n a ij U j (3.45) j= and so a ij U j is the effect of the species U j on the growth of U i.ifa ij > 0 then U j directly enhances U i s growth while if a ij < 0 it diminishes it. If a ij > 0anda ji > 0 then U i and U j enhance each other s growth and so they are in a symbiotic interaction. If a ij < 0anda ji < 0 then they are in competition. May (975) gives a survey of some

25 3.7 General Models and Cautionary Remarks 03 generalised models and, in his discussion on stability versus complexity, gives some results for stability based on properties of the community matrix. There has been a considerable amount of study of systems where the community matrix has diagonal symmetry or antisymmetry or has other rather special properties, where general results can be given about the eigenvalues and hence the stability of the steady states. This has had very limited practical value since models of real situations do not have such simple properties. The stochastic element in assessing parameters mitigates against even approximations by such models. However, just as the classical Lotka Volterra system is not relevant to the real world, these special models have often made people ask the right questions. Even so, a preoccupation with such models or their generalizations must be avoided if the basic aim is to understand the real world. An important class of models which we have not discussed is interaction models with delay. If the species exhibit different or distributed delays, such models open up a veritable Pandora s box of solution behaviour which to a large extent is still relatively unexplored. If we consider three or more species, aperiodic behaviour can arise. Lorenz (963) first demonstrated this with the model system du = a(v u), dv = uw + bu v, dw = uv cw, where a, b and c are positive parameters. (The equations arose in a fluid flow model.) As the parameters are varied the solutions exhibit period doubling and eventually chaos or aperiodicity. Many authors have considered such systems. For example, Rössler (976a,b, 979, 983), Sparrow (982, 986) and Strogatz (994) have made a particular study of such systems and discovered several other basic examples which show similar properties; see also the book edited by Holden (986). It would be surprising if certain population interaction models of three or more species did not display similar properties. Competition models of three or more species produce some unexpected results. Evidence for chaos (even complex oscillations) in wild populations is difficult to find as well as difficult to verify unequivocably. It has been suggested therefore that evolution tends to preserve populations from such chaotic behaviour. Possible chaotic population dynamics which results from natural selection has been investigated in an interesting article by Ferrière and Gatto (993). From their results they hypothesize that evolution might support considerably more chaotic population dynamics than believed up to now. Controversially they suggest that chaos is possibly optimal in population dynamics. They suggest, in fact, that chaos could be an optimal behaviour for various biological systems. This conclusion is in line with the views expressed by Schaffer and Kot (986) with regard to epidemics. Notwithstanding the above, evolutionary development of complex population interactions seems to have generally produced reasonably stable systems. From our study of interaction models up to now we know that a system can be driven unstable if certain parameters are changed appropriately, that is, pass through bifurcation values. It should therefore be a matter of considerable scientific study before any system is altered by external manipulation. The use of models to study the effect of artificially interfering in such trophic webs is essential and can be extremely illuminating. Had this been done it