Quantitative Methods for Ecology and Evolutionary Biology (Cambridge, 2006) - Chapter 4 pot

Chapter The evolutionary ecology of parasitoids Insect parasitoids – those insects that deposit their eggs on or in the eggs, larvae or adults of other insects and whose offspring use the resources of those hosts to fuel development – provide a rich area of study for theoretical and mathematical biology They also provide a broad collection of examples of how the tools developed in the previous chapters can be used (and they are some of my personally favorite study species; the pictures shown in Figure 4.1 should help you see why) There is also a rich body of experimental and theoretical work on parasitoids, some of which I will point you towards as we discuss different questions The excellent books by Godfray (1994), Hassell (2000a), and Hochberg and Ives (2000) contain elaborations of some of the material that we consider These are well worth owning Hassell (2000b), which is available at JSTOR, should also be in everyone’s library It is helpful to think about a dichotomous classification scheme for parasitoids using population, behavioral, and physiological criteria (Figure 4.2) First, parasitoids may have one generation (univoltine) or more than one generation (multivoltine) per calendar year Second, females may lay one egg (solitary) or more than one egg (gregarious) in hosts Third, females may be born with essentially all of their eggs (pro-ovigenic) or may mature eggs (synovigenic) throughout their lives (Flanders 1950, Heimpel and Rosenheim 1998, Jervis et al 2001) Each dichotomous choice leads to a different kind of life history 133 134 The evolutionary ecology of parasitoids (a) (b) (c) (d) (e) Figure 4.1 Some insect parasitoids and insects that have life histories that are similar to parasitoids (a) Halticoptera rosae, parasitoid of the rose hip fly Rhagoletis basiola, (b) Aphytis lingannensis, parasitoid of scale insects, and (c) Leptopilinia heterotoma, parasitoid of Drosophila subobscura (d, e) Tephritid (true) fruit flies have life styles that are parasitoid-like: adults are free living, but lay their eggs in healthy fruit The larvae use the resources of the fruit for development, then drill a hole out of the fruit and burrow into the ground for pupation Here I show a female rose hip fly R basiola (d) ovipositing, and two males of the walnut husk fly R compleata (e) fighting for an oviposition site (the successful male will then try to mate with females when they come to use the oviposition site) The black trail under the skin of the walnut is the result of a larva crawling about and creating damage between the husk and the shell as it uses the resource of the fruit The Nicholson–Bailey model and its generalizations (a) Generations per year: Multivoltine (b) Eggs per host: >1 Univoltine Figure 4.2 A method of classifying parasitoid life histories according to population, behavioral and physiological criteria >1 Solitary Gregarious (c) Egg production after emergence: (d) Combining the characteristics: Multivoltine =0 Univoltine >0 Solitary Gregarious Pro-ovigenic Synovigenic Pro-ovigenic Synovigenic The Nicholson–Bailey model and its generalizations The starting point for our (and most other) analysis of host–parasitoid dynamics is the Nicholson–Bailey model (Nicholson 1933, Nicholson and Bailey 1935) for a solitary univoltine parasitoid We envision that hosts are also univoltine, in a season of unit length, in which time is measured discretely and in which H(t) and P(t) denote the host and parasitoid populations at the start of season t Each host that survives to the end of the season produces R hosts next year The parasitoids search randomly for hosts, with search parameter a, so that the probability that a single host escapes parasitism from a single parasitoid is eÀa Thus, the probability that a host escapes parasitism when there are P(t) parasitoids present at the start of the season is eÀaP(t) These absolutely sensible assumptions lead to the dynamical system Ht ỵ 1ị ẳ RHtịeaPtị Pt ỵ 1ị ẳ Htị1 eaPtị ị (4:1) Note that in this case the only regulation of the host population is by the parasitoid Hassell (2000a, Table 2.1) gives a list of 11 other sensible assumptions that lead to different formulations of the dynamics The first question we might ask concerns the steady state of Eq (4.1), obtained by assuming that H(t ỵ 1) ẳ H(t) and P(t ỵ 1) ẳ P(t) These are easy to find Exercise 4.1 (E) Show that the steady states of Eqs (4.1) are " P ẳ logRị a " Hẳ R logRị aR 1Þ 135 (4:2) 136 The evolutionary ecology of parasitoids (a) (b) 100 300 90 Hosts Hosts 250 80 Parasitoids Population size 60 50 40 Parasitoids 30 20 200 150 100 50 10 10 15 Generation 20 25 30 10 15 Generation 20 (c) 60 60 40 30 20 10 Hosts 50 50 Population size 40 Parasitoids 30 20 10 10 20 30 40 50 60 Week 10 Week 15 (d) 500 Hosts 400 Population size Hosts Population size 70 300 200 Parasitoids 100 0 20 40 60 80 Week 100 120 140 160 20 25 30 The Nicholson–Bailey model and its generalizations which shows that R > is required for a steady state (as it must be) and that higher values of the search effectiveness reduce both host and parasitoid steady state values The sad fact, however, is that this perfectly sensible model gives perfectly nonsensical predictions when the equations are iterated forward (Figure 4.3): regardless of parameters, the model predicts increasingly wild oscillations of population size until either the parasitoid becomes extinct, after which the host population is not regulated, or both host and parasitoid become extinct To be sure, this sometimes happens in nature, usually this is not the situation Instead, hosts and parasitoids coexist with either relative stable cycles or a stable equilibrium In a situation such as this one, one can either give up on the theory or try to fix it My grade PE teacher, Coach Melvin Edwards, taught us that ‘‘quitters never win and winners never quit,’’ so we are not going to give up on the theory, but we are going to fix it The plan is this: for the rest of this section, we shall explore the origins of the problem In the next section, we shall fix it As a warm-up, let us consider a discrete-time dynamical system of the form N t ỵ 1ị ẳ f ðN ðtÞÞ 137 N f (N ) 45 ο N N Figure 4.4 The stability of the steady state of the one dimensional dynamical system N(t ỵ 1) ẳ f(N(t)) is determined by the derivative of f(N) " evaluated at the steady state N (4:3) where f (N) is assumed to be shaped as in Figure 4.4, so that there is a " " " steady state N defined by the condition N ẳ f N ị To study the stability " ỵ ntị where n(t), the perturbaof this steady state, we write N tị ẳ N tion from the steady state, is assumed to start off small, so that " jnð0Þj ( N We then evaluate the dynamics of n(t) from Eq (4.3) by Taylor expansion of the right hand side keeping only the linear term  df   ntị " " " N ỵ nt ỵ 1ị ẳ f N ỵ ntịị % f N ị þ dN N " (4:4) Figure 4.3 Although the Nicholson–Bailey model seems to be built on quite sensible assumptions, its predictions are that host and parasitoid population sizes will oscillate wildly until either the parasitoids become extinct (panel a, H(1) ¼ 25, P(1) ¼ 8, R ¼ and a ¼ 0.06) and the host population then grows without bound, or the hosts become extinct (panel b, H(1) ¼ 25, P(1) ¼ 8, R ¼ 1.8 and a ¼ 0.06), after which the parasitoids must become extinct (c) Some host–parasitoid systems exhibit this kind of behavior On the left hand side, I show the population dynamics of the bruchid beetle Callosobruchus chinesis in the absence of a parasitoid (note that this really cannot match the assumptions of the Nicholson–Bailey model, because there is regulation of the population in the absence of the parasitoid); on the right hand side, I show the beetle and its parasitoid Anisopteromalus calandre In this case, the cycles are indeed very short (d) On the other hand, many host–parasitoid systems not exhibit wild oscillations and extinction Here I show the dynamics of laboratory populations of Drosophila subobscura and its parasitoid Asobara tabida The data for panels (c) and (d) are compliments of Dr Michael Bonsall, University of Oxford Also see Bonsall and Hastings (2004) 138 The evolutionary ecology of parasitoids " " Since N ¼ f ðN Þ and setting f N ¼ df =dN jN we conclude that n(t) " approximately satisfies nðt ỵ 1ị ẳ f N ntị (4:5) and we conclude that the steady state will be stable, in the sense that " perturbations from it decay, if j f N ðN Þj51 Exercise 4.2 (E) For more practice determining when a steady state is stable, the computation for the discrete Ricker map  ! Ntị N t ỵ 1ị ¼ NðtÞexp r À K and show that the condition is |1 À r| < 1, or < r < But we have a two dimensional dynamical system Since what follows is going to be a lot of work, we will the analysis for the more general host–parasitoid dynamics Basically, we for the steady state of a two dimensional discrete dynamical system the same kind of analysis that we did for the two dimensional system of ordinary differential equations in Chapter Because the procedure is similar, I will move along slightly faster (that is, skip a few more steps) than we did in Chapter Our starting point is Ht ỵ 1ị ẳ RHtị f Htị; Ptịị Pt ỵ 1ị ẳ Htị1 f ðHðtÞ; PðtÞÞÞ (4:6) " " which we assume has a steady state H; Pị We now assume that " ỵ htị and Ptị ẳ P ỵ ptị, substitute back into Eq (4.6), " Htị ẳ H Taylor expand keeping only linear terms and use o(h(t), p(t)) to represent terms that are higher order in h(t), p(t), or their product to obtain " " " " H ỵ ht ỵ 1ị ẳ RH ỵ htịịẵ f H; Pị ỵ f H htị ỵ f P ptị ỵ ohtị; ptịị " " " " P ỵ pt ỵ 1ị ẳ H ỵ htịịẵ1 f H; Pị f H htị f P ptị ỵ ohtị; ptịị (4:7) where f H ¼ ðq=qHÞf ðH; PÞjðH;PÞ and fP is defined analogously " " Now, from the definition of the steady states we know that " " " " " " H ¼ RHf ðH; PÞ, which also means that Rf ðH; PÞ ¼ 1, and that " ¼ Hð1 À f ðH; PÞÞ We now use these last observations concerning " " " P the steady state as we multiply through, collect terms, and simplify to obtain The Nicholson–Bailey model and its generalizations " " ht ỵ 1ị ẳ htị1 ỵ RHf H ị ỵ RHf P ptị ỵ ohtị; ptịị " " pt ỵ 1ị ẳ htị1 1=Rị Hf H ị Hf P ptị ỵ ohtị; ptịị (4:8) Unless you are really smart (probably too smart to find this book of any use to you), these equations should not be immediately obvious On the other hand, you should be able to derive them from Eqs (4.7), with the intermediate clues about properties of the steady states in about 3–4 lines of analysis for each line in Eqs (4.8) If we ignore all but the linear terms in Eqs (4.8) we have the linear system ht ỵ 1ị ẳ ahtị ỵ bptị pt ỵ 1ị ẳ chtị ỵ dptị (4:9) with the coefficients a, b, c, and d suitably defined; as before, we can show that this is the same as the single equation ht ỵ 2ị ẳ a þ dÞhðt þ 1Þ þ ðbc À adÞhðtÞ (4:10) by writing h(t ỵ 2) ẳ ah(t ỵ 1) ỵ bp(t þ 1), p(t þ 1) ¼ ch(t) þ dp(t) ¼ ch(t) ỵ (d/b)(h(t ỵ 1) ah(t)) and simplifying (Once again you should not necessarily see how to this in your head, but writing it out should make things obvious quickly.) If we now assume that h(t) $ lt (there is actually a constant in front of the right hand side, as in Chapter 2, but also as before it cancels), we obtain a quadratic equation for l: l2 a ỵ dịl ỵ ad bc ẳ (4:11) which I am going to write as l2 À l ỵ ẳ with the obvious identification of the coefficients Also as before, Eq (4.11) will have two pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi roots, which we ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p will denote by l1 ẳ =2ị ỵ =2ị and À =2Þ The steady state will be stable if perturl2 ¼ ð =2j > À =2; if we square both sides of this expression the condition becomes | | ỵ ( 2/4) > ( 2/4) and this simplifies to ỵ > | | Our first condition was ! and we have agreed that | | < so that < Therefore, > ! 4 , so that > or > ỵ When we combine the two conditions, we obtain the criterion for stability that (Edelstein-Keshet 1988) > ỵ > j j (4:12) Hassell (2000a) gives (his Eqs (2.2), (2.3)) the application of this condition to the general Eqs (4.6) In the case of Nicholson–Bailey dynamics, f (H, P) ¼ exp(ÀaP), so that fH ¼ and fP ¼ Àaexp(ÀaP); these need to be evaluated at the steady states and the coefficients a, b, c, and d in Eq (4.9) evaluated so that we can then determine and Exercise 4.3 (M/H) For NicholsonBailey dynamics show that ẳ ỵ [log(R)/(R 1)] and that ¼ Rlog(R)/(R À 1) Then show that since R > 1, ỵ > However, also show that ỵ > by showing that > (to this, consider the function g(R) ¼ Rlog(R) À R ỵ for which g(1) ẳ and show that g0 (R) > for R > 1) thus violating the condition in Eq (4.12), and thus conclude that the Nicholson– Bailey dynamics are always unstable What biological intuition underlies the instability of the Nicholson–Bailey model? There are two answers First, the per capita search rate of the parasitoids is independent of population size of parasitoids (which are likely to experience interference when population is high) Second, there is no refuge for hosts at low density – the fraction of hosts killed depends only upon the parasitoids and is independent of the number of hosts We now explore ways of stabilizing the Nicholson–Bailey model Stabilization of the Nicholson–Bailey model Stabilization of the Nicholson–Bailey model I now describe two methods that are used to stabilize Nicholson–Bailey population dynamics, in the sense that the unbounded oscillations disappear Note that we implicitly define that a system that oscillates but stays within bounds is stable (Murdoch, 1994) The methods of stabilization rely on variation and refuges Variation in attack rate The classic (Anderson and May 1978) means of stabilizing the Nicholson–Bailey model is to recognize that not all hosts are equally susceptible to attack, for one reason or another To account for this variability, we replace the attack rate a by a random variable A, with E{A} ¼ a, so that the fraction of hosts escaping attack is exp(ÀAP) However, to maintain a deterministic model, we average over the distribution of A; formally Eq (4.1) becomes Ht ỵ 1ị ẳ RHtịEA feAPtị g Pt ỵ 1ị ẳ Htị1 EA feAPtị gị (4:13) where EA{ } denotes the average over the distribution of A For the distribution of A, we choose a gamma density with parameters and k We then know from Chapter that the resulting average of exp(ÀAP(t)) will be the zero term of a negative binomial distribution, so that EA feAPtị g ẳ  k ỵP (4:14) Since the mean of a gamma density with parameters and k is k/, it would be sensible for this to be the average value of the attack rate so that a ¼ k/; we choose ¼ k/a We then multiply top and bottom of the right hand side of Eq (4.14) by k/ to obtain !k k k ỵ aPtị  h ik  k Pt ỵ 1ị ẳ Htị kỵaPtị Ht ỵ 1ị ẳ RHtị (4:15) This modification of the Nicholson–Bailey model is sufficient to stabilize the population dynamics (Figure 4.6) To help understand the intuition that lies behind this stabilization, I note the following remarkable feature (Pacala et al 1990): the stabilization occurs as long as the overdispersion parameter k < I have illustrated this point in 141 ... P PðtÞ dt (4: 24) 147 148 The evolutionary ecology of parasitoids Equations (4. 19)– (4. 24) constitute the description of a host parasitoid system with overlapping generations and potentially different... graph with l on the x-axis and y ¼ l or y ¼ ÀreÀl on the y-axis and look for their intersections and note that if  ¼ then l ¼ Àr) More advanced models for population dynamics 149 To further increase... fH ¼ and fP ¼ Àaexp(ÀaP); these need to be evaluated at the steady states and the coefficients a, b, c, and d in Eq (4. 9) evaluated so that we can then determine and Exercise 4. 3 (M/H) For NicholsonBailey