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1 1 The effect of adaptive change in the prey on the dynamics of an 2 exploited predator population 3 4 5 Peter A. Abrams, Department of Zoology, University of Toronto, 25 Harbord Street, 6 Toronto, Ontario M5S 3G5, Canada (phone 416-978-1014; fax 416-978-8532; email 7 abrams@zoo.utoronto.ca) 8 9 10 11 12 Hiroyuki Matsuda, Faculty of Environment & Information Sciences, Yokohama National University , 79-7, Tokiwadai, Hodogaya-ku, Yokohama, Kanagawa 240-8501, Japan 2 13 Abstract 14 Simple mathematical models are analyzed to determine the relationship between 15 harvesting effort and stock size for a predator in a system in which the prey adapt to the 16 risk of predation. Two types of model are studied: in the first, the prey has a tradeoff 17 between increasing its own net reproductive rate and increasing its vulnerability to the 18 predator. In the second class of models, there are two prey species that differ in their 19 vulnerability to the predator. Each prey species has fixed, non-evolving characteristics, 20 but changes in the average characteristics within the prey trophic level can occur via 21 shifts in the relative abundance of the two species. In both classes of models, the 22 equilibrium predator population can increase with increasing harvest of the predator. In 23 the case of the 2-prey model, the predator's equilibrium population always increases with 24 an increased harvest rate if the two prey coexist and share a single resource. The 25 predator's equilibrium population often decreases from its maximum size to zero over a 26 very small range of harvest rates, once those rates become high enough. Because 27 increased stock size is often used to justify increased harvest rates, this type of 28 relationship poses a risk that harvest rate will increase to the point where the stock 29 quickly collapses. The results are relevant to understanding and predicting the changes in 30 population size of species experiencing declining environmental conditions. 31 32 33 Keywords: food web, predator-prey system, prey adaptation, sustainable yield 3 34 Fisheries models have largely employed a single-species perspective. 35 Multispecies models (reviewed by Hollowed et al. 2000; Latour et al. 2003) have 36 generally assumed that the characteristics of individual species do not change through 37 time. Although one popular simulation package (Ecosim; Walters et al. 2000) 38 incorporates both a multi-species approach and an assumption of transitions from 39 vulnerable to invulnerable states, we currently lack a good understanding of the 40 circumstances when a 2- or more-species system incorporating adaptive evolution or 41 behaviour will exhibit dynamics that are fundamentally different from those of single 42 species systems. Understanding the properties of models with small numbers of species 43 is important in determining what factors should be included in larger scale simulation 44 models, and in understanding the dynamics of those larger models. 45 An earlier article (Matsuda and Abrams 2004) described how population cycles 46 and/or adaptive change in the predator could qualitatively change the relationship 47 between harvesting effort applied to the predator and predator population size. One of 48 the main results of that analysis was that, unlike single-species models, stock size could 49 increase as harvest increased in both stable and unstable predator-prey systems. The 50 present article will extend the previous one to examine the effect of adaptation in the prey 51 population on the harvesting effort – stock size relationship for the predator. Studies of 52 several fish species have shown that prey often exhibit costly shifts in either foraging 53 behaviour (Turner and Mittelbach 1990; Fraser and Gilliam 1992; Eklöv and Persson 54 1995), or genetically determined life history traits (Reznick and Bryga 1990) in response 55 to greater abundance of piscivorous predators. The foraging shifts are examples of life 56 history plasticity, since they involve increased survival at the expense of decreased 4 57 growth or reproduction. There is mounting evidence that adaptive changes, both 58 behavioural and evolutionary, affect the dynamics of exploited fish populations (Heino 59 and Godø 2002). Here we take a general theoretical approach to determine the potential 60 population dynamical consequences of prey adaptation in systems in which the predator 61 is exploited. 62 We begin by noting some of the properties of single-species models in which the 63 harvested stock has a relatively homogeneous population whose members have fixed 64 characteristics, and whose limiting dynamic in the absence of environmental variability is 65 a stable equilibrium. Such a population exhibits a decrease in population size in response 66 to an increased per capita harvest rate. The initial and ultimate responses to increased 67 harvest are both negative in sign. In such systems, feedback control of harvest effort 68 represents a safe method for maintaining a healthy stock size, provided stock size can be 69 measured, fishing effort can be controlled, and the target stock size is not too small. 70 These results are straightforward conclusions of the models presented in standard 71 fisheries texts (e.g., Hilborn and Walters 1992; Quinn and DeRiso 1999). The remainder 72 of this article will show how adaptive change in the vulnerability of a prey species to a 73 predator can alter these apparently intuitive conclusions. 74 75 The relationship between predator population size and predator 76 mortality in a predator-prey system without prey adaptation 77 The analysis here assumes a simple predator-prey model in which the populations 78 are not structured. The prey has density dependent population growth, with population 79 size R, and per capita growth rate f(R). An average predator individual consumes prey at 5 80 a rate given by CRg(CR), where C is a per capita attack rate, and g(CR) is a decreasing 81 function (the 'satiation function') describing the proportional reduction in the predator's 82 consumption rate as the encounter rate with prey increases; this may occur because of 83 handling time or other factors. We assume the functional response, CRg(CR), increases 84 with R. The consumed prey is converted into new predators according to a birth rate 85 function, b, which is an increasing function of food intake rate (i.e., the functional 86 response). The per capita death rate of predators is given by d. If predator population 87 size is denoted N, these assumptions result in the following population dynamical model: 88 89 dR/dt = R[f(R) – CNg(CR)] (1a) 90 dN/dt = N[b(CRg(CR)) – d] (1b) 91 92 The equilibrium predator abundance, N*, is determined by eq (1a), and is simply 93 94 N* = f(R)/[Cg(CR)] (2) 95 96 Increased mortality (d) of the predator is expected to increase the equilibrium resource 97 population based on the equilibrium condition for eq (1b). A greater R decreases the 98 resource per capita growth rate, and increases predator satiation, which means a smaller 99 g. Thus, the equilibrium predator population may increase with its own mortality if g 100 decreases more rapidly than f as the result of its mortality. This is possible when the 101 predator has a saturating (i.e., Holling type 2 or 3) functional response. However, as is 102 well-known (Rosenzweig and MacArthur 1963), the fact that N* increases with d implies 6 103 that the vertical predator isocline intersects the prey isocline where the latter has a 104 positive slope. This in turn means that the equilibrium is an unstable focus, and that 105 cycles will occur. Numerical results show that, although the mean predator population 106 size differs from N*, the mean also increases with the predator's own mortality (Abrams 107 et al 1997; Abrams 2002). 108 Thus, the conclusion from simple predator-prey models with fixed prey 109 characteristics is that the predator population will only increase with its own mortality 110 rate when there are cycles. This conclusion must be modified if the predator's per capita 111 growth rate is directly and negatively affected by the predator population size; e.g., if d is 112 replaced by d0 + I(N), where d0 is a constant and I is an increasing function of predator 113 population size. In this case it is possible for the equilibrium predator population, N*, to 114 increase with the predator’s death rate, d0 in a stable system, although this typically 115 requires rather careful balancing of parameter values, and only occurs over a narrow 116 range of d0. An increase in fishing mortality seems most likely to affect density 117 independent mortality, d0, rather than changing the density dependent death rate, I(N). 118 119 The effect of adaptive change of prey vulnerability on the relationship 120 between predator population size and predator mortality 121 There are several ways in which adaptive change in prey characteristics can alter 122 the manner in which the predator's population size changes with its own mortality. Prey 123 vulnerability helps determine the attack rate, C, and changes in C are expected to have 124 correlated effects on prey growth rate (Lima 1998). The per capita growth function of 125 prey is assumed to increase with C and to decrease with R. Therefore, f(R) in eqs (1, 2) 7 126 must be replaced by f(R, C). If f increases due to an increase in C following greater 127 predator mortality, equation (2) shows that this could cause an increase in N* with 128 increased d. In addition, an increase in both C and R following greater predator mortality 129 can decrease the function, g, which also increases N* (see eq (2)). The increased 130 predator satiation (smaller g) also makes it adaptive for the prey to reduce costly 131 defences, again implying an increase in both C and the per capita growth, f. The 132 remainder of this section develops this qualitative argument by augmenting the 133 population dynamics model (eqs. (1)) with an equation for adaptive change in the value 134 of C, assuming that lower C implies lower f. We assume that this trait is subject to 135 stabilizing selection in the absence of the predator, so that 2f/C2 < 0. 136 We assume that the prey's per capita growth rate f is an increasing function of 137 vulnerability, C at values of C near the evolutionary equilibrium, implying a cost to 138 defense. A model that describes adaptive change in an approximate manner for many 139 circumstances (Abrams et al. 1993; Abrams 2001) assumes that the rate of change of a 140 trait (e.g., C) in the population is proportional to the rate of change of a particular 141 individual's fitness with its own trait value, evaluated at the population mean. This 142 corresponds to the general finding that behaviour changes more rapidly when the reward 143 from a unit change is greatest. Similarly, genetically determined traits evolve most 144 rapidly when the slope of the relationship between the trait and fitness is most steep. For 145 prey adaptation in the context of Equations (1a,b), the vulnerability trait changes 146 according to: 147 148 dC/dt = v[f/C – Ng(CR)] (3) 8 149 150 where v is an adaptive rate constant that scales the rate of change in the trait relative to 151 the rate of change of population densities, and the expression in brackets is the derivative 152 of an individual’s fitness with respect to its trait, C. The derivative of individual fitness 153 with respect to the trait determines the rate of change in the mean value of the trait in 154 models of mutation-limited evolution (Dieckmann and Law 1996), quantitative genetic 155 models (Abrams et al. 1993; Abrams and Matsuda 1997), and behavioural models 156 (Taylor and Day 1997; Abrams 2001). In all of these models, equation (3) is an 157 approximation based on the assumption that the amount of variation in the trait C is 158 sufficiently small. If the change is genetic, v is the additive genetic variance of the trait. 159 If genetic or behavioural variation is depleted when C approaches extreme values, v will 160 be a function of C (Abrams 2001). 161 The system consisting of eqs (1a,b) and eq (3) can be analyzed without adopting 162 specific functional forms for f and g. It is possible to determine when the equilibrium 163 will be locally stable, and to determine how the equilibrium values of the variables 164 change with the predator mortality, d. The details of this analysis are provided in the 165 appendix. The condition required for N* to increase with d is that the following 166 derivative be positive: 167 168 169 2 2 f f 2 f f 2 f Ng ' C R C 2 C 2 C R N * C R R d 2 f f 2 f gb ' g CRg ' C 2 R C 2 C R R C (4) 9 170 The appendix discusses the signs of the terms in this expression. If the equilibrium of the 171 system is locally stable, both the quantity in square brackets in the denominator, and the 172 entire denominator must be negative (see appendix). In addition we assume that f/R < 173 0, because the prey’s per capita growth rate declines with its own population size, and 174 2f/C2 < 0, as mentioned above. Furthermore, g + CRg' > 0, because the functional 175 response increases with CR (i.e., it increases with the availability of prey). These 176 considerations mean that, if there is no satiation (g' = 0), the numerator of eq (4) must be 177 positive, eq (4) must be negative, and the predator must therefore decrease in population 178 size with increases in its own death rate. However, with satiation (g' < 0) the second of 179 the two terms in the numerator of eq (4), having the form Ng'[…], is usually negative, 180 making it possible for the predator to increase as its own mortality increases. One factor 181 that contributes to an increase in predator density in the increase in prey growth, due to a 182 larger vulnerability, C. The appendix shows that C*/d has the sign of –[f/R – 183 C2f/(CR)]. Thus, unless there is a large negative interaction of the effects of C and R 184 on prey growth, C* will increase with d. The greater per-individual productivity of prey 185 provides the basis for the predator population to increase. 186 Some of the necessary conditions for stability of eqs (1a, 1b, 3) have a simple 187 form. Unfortunately, the full necessary and sufficient local stability conditions for the 188 system consisting of eqs (1a, 1b, 3) is sufficiently complicated to be of little use in 189 understanding stability, unless specific forms are adopted for the functions in the model. 190 Here we simply examine stability numerically using specific forms for the three 191 unspecified functions in eqs. (1a, 1b, 3). We assume: 192 10 193 f(R,C) = rm + r1C – r2C2 – (R/K) (5a) 194 g(CR) = 1/(1 + hCR) (5b) 195 b(CRg(CR)) = b0CRg(CR) (5c) 196 197 Here, the prey’s per capita growth rate, f, changes in a unimodal fashion with the trait, C, 198 and has a minimum value of rm. Equation (5b) implies that the predator has a Holling 199 (1959) disk equation (type 2) functional response, with handling time h. Equation (5c) 200 means that the birth rate function is linear; births are directly proportional to food intake 201 with a proportionality constant, b0. In the absence of the predator, the prey fitness is 202 given by f, and there is quadratic stabilizing selection on the prey’s trait, C. The 203 functions given by eqs (5) also permit an analytical solution for the equilibrium R, N, and 204 C, although the formulas for the equilibrium values are quite lengthy. The equilibrium 205 predator population is plotted as a function of predator death rate for a particular set of 206 parameters in figure 1; it is clear that the equilibrium predator population increases with 207 mortality over the vast majority of the range of potential mortalities. The equilibrium C 208 increases at an accelerating rate over the same range of mortalities in this example. The 209 figure legend discusses the stability of these equilibrium values. This model is similar to 210 a model whose dynamics were studied in Abrams and Matsuda (1997), and there are 211 often alternative attractors (see legend of fig. 1). Because the trait dynamics given by eq 212 (3) alone can result in biologically unrealistic (either very large or negative) values of the 213 trait, the numerical results assume that there is selection for less extreme values if C 214 approaches zero or a maximum value (see legend of fig. 1). 11 215 It is important to note that the prediction of an increased predator population is 216 based on the change in equilibrium population size. The initial response of the predator 217 to an increase in its mortality is always a decrease in population size. This is followed by 218 an increase in both the prey's abundance and its trait value, which eventually allow the 219 predator to increase above its initial population size. Figure 2 shows an example of the 220 transient responses of both densities and traits to a doubling of the predator's per capita 221 mortality rate for the system described in figure 1. 222 223 The effect of adaptive change in the prey community composition on the 224 relationship between predator population size and predator mortality 225 A predator-prey system that contains more than a single prey species can exhibit 226 adaptive change at the level of the prey assemblage, even when no species within that 227 assemblage undergoes evolutionary change. Abrams and Matsuda (1997) showed that a 228 system with two competing prey that differed in their growth rates and predator 229 vulnerabilities could exhibit dynamics very similar to those of a single-prey model in 230 which the single prey species had an adaptive tradeoff between growth rates and predator 231 vulnerability. This section will examine whether and when such systems can exhibit the 232 type of increase of predator abundance with predator harvest demonstrated in the 233 previous section. 234 The first model considered here consists of two prey species with a single shared 235 resource as well as a single shared predator. This 'diamond food web' has been analyzed 236 in different ways by various authors (e.g. Armstrong 1979; Holt et al. 1994; Leibold 237 1996). Neither these studies, nor more recent ones (Grover and Holt 1998; Abrams 12 238 1999), have pointed out the implications of the food web structure for the response of the 239 predator to increased mortality. If the predator has linear functional responses to both 240 prey, and the prey do not have any direct effects of population size on their own growth 241 rate, then the predator's equilibrium population is independent of its own mortality; this 242 was noted by Leibold (1996) for a similar model of the diamond web, and is a 243 consequence of the fact that the more vulnerable of the two prey types increases at the 244 expense of the less vulnerable prey when mortality is imposed on the predator. When the 245 predator's functional response is nonlinear, its population always increases with an 246 increase in its own mortality as long as the two prey are able to coexist. 247 Although we have assumed that there is no adaptive change in either C1 or C2, the 248 relative frequencies of the prey change when mortality is imposed on the predator. The 249 resulting increase in the more vulnerable prey increases predator satiation, reducing the 250 effective attack rates on both prey. It is these reduced attack rates that allow the predator 251 to increase in abundance. These results can be demonstrated by an analysis of a general 252 model with three trophic levels. Having three levels entails a shift in notation; we now 253 denoted the predator population size by P, the two prey populations by Ni, and the 254 resource population by R. This yields a model having the following form: 255 256 dR/dt = R[f(R) – C1N1 – C2N2] (6a) 257 dN1/dt = N1[b1(C1R) – d1 – s1PG(s1N1+es2N2)] (6b) 258 dN2/dt = N2[b2(C2R) – d2 – s2PG(s1N1+es2N2)] (6c) 259 dP/dt = P[B{(s1N1 + es2N2)G(s1N1+es2N2)} – D] (6d) 260 13 261 The symbols that are common to this system and eqs (1) have the same meaning. The 262 predator’s satiation function is denoted G. The parameter e measures the energetic 263 content (or nutritional value) of prey 2 relative to prey 1, and it is assumed that energetic 264 content (or nutritional value) of prey determines satiation. N1 and N2 represent the 265 populations of the two prey species. The per capita death rates of the two prey and the 266 predator are di and D respectively, while the conversion functions of resource to prey and 267 prey to predator are bi and B respectively. Note that the parameters of both prey are 268 constant; adaptive change can only occur at the prey trophic level by shifts in the 269 abundances of the two species. The two prey species are assumed to have linear 270 functional responses to the resource to simplify the algebra and to reduce the possibility 271 of complex dynamics. The same techniques described in the appendix for eqs (1a, 1b, 3) 272 can be applied to this system, and the resulting expression for the change in P* with D is, 273 274 P * PG ' D B ' G G G '(s1 N1 es2 N 2 ) (7) 275 276 This expression must be positive because G' is negative and B' positive by definition, and 277 the quantity in parentheses in the denominator is the derivative of the predator's 278 functional response with respect to food intake, which must also be positive at a stable 279 equilibrium. It should be noted that a sufficiently large magnitude increase or decrease in 280 D is likely to eliminate one of the prey species, and further changes in D will only have 281 the intuitive effects on the average value of P (i.e., higher D means smaller P) if the 282 system is stable. Note also that without predator satiation (G = 1; G' = 0), the predator's 283 equilibrium abundance does not change with an increase in its own per capita mortality. 14 284 Thus, the harvesting rate of predators would not change the equilibrium abundance as 285 long as both prey are present. 286 The local stability conditions of an equilibrium of eqs (6) having all four species, 287 are quite complicated. However, the equilibrium must be stable when there is no 288 satiation (G = 1), since the Jacobian matrix then satisfies qualitative stability conditions 289 (Jefferies 1974). Simulations have shown that a wide range of parameters will produce a 290 stable system, as long as the predator's death rate is not too low and the satiation function 291 is not too strongly concave. 292 The quantitative change in predator population size with its own mortality can be 293 illustrated for a particular version of the diamond web model. Here, we introduce some 294 simplifications of eqs (6) that allow the equilibrium predator population size to be 295 determined analytically. The numerical response functions, bi and B in eqs (6) are all 296 linear, resource growth is logistic (f(R) = r(1 – (R/K))), and the satiation function has the 297 form of Holling's (1959) disk equation; G(x) = 1/(1 + hx). In addition, we assume that 298 the resource has sufficiently rapid dynamics relative to the prey that it can be assumed to 299 always have its steady-state abundance of (K/r)(r – C1N1 – C2N2). This allows the 300 dynamics of the two prey to be reformulated as Lotka-Volterra competition equations, 301 with the product of the two competition coefficients being unity (see Abrams 1999). We 302 also assume that s2 = 0; i.e., prey 2 is immune to predation. Thus, the model becomes: 303 304 dN1/dt = r1N1[1 – (N1 + 12N2) / K1] – s1N1P/(1+hs1N1) (8a) 305 dN2/dt = r2N2[1 – (N2 + 21N1) / K2] (8b) 306 dP/dt = P[(B0s1N1) / (1+hs1N1) – D] (8c) 15 307 where B0 is a constant, ij is the competition coefficient of prey j on prey i, and the 308 parameters ri and Ki now refer to the prey rather than the resource. Equations (8) have a 309 relatively simple analytical expression for the equilibrium predator population: 310 311 P* K1s12 B0 Dh if D 312 313 B0 r1 B0 s1 K1 12 K 2 D 1 s1hK1 12 21 s1hK 2 P* 2 K 2 s1 B0 21 K 2 s1h K 2 s1 B0 B0 r D 1 if D 21 K 2 s1h s1 B0 Dh s1 K1 B0 Dh (9a) (9b) 314 315 where the second solution corresponds to the case where prey 2 is excluded. Figure 3 316 plots eq (9a,b) as a function of the predator’s per capita death rate, D; the different lines 317 are characterized by different relative carrying capacities of the two prey. As can be seen 318 from the figure, the greatest proportional increase in predator population over the widest 319 range of the potential mortality rates occurs when the invulnerable 'prey' species has only 320 a slightly smaller carrying capacity (i.e., a slight competitive disadvantage) relative to the 321 prey species that is vulnerable to the competitor. However, the predator is relatively rare 322 under these circumstances. The abrupt change in slope of the relationships shown in fig. 323 3 occurs when the predator mortality is such that the less vulnerable prey is excluded by 324 the more vulnerable prey; at still higher mortalities, the equilibrium predator population 325 decreases. 326 327 The above results suggest that a slightly more complex foodweb might be capable of producing similar phenomena. In particular, it is of interest to determine how strong 16 328 the competition between the two prey must be in order to produce the result we have 329 stressed above; the predator's population size increases as its own per capita mortality 330 increases. This can be investigated by analyzing a model similar to eqs (6), but with a 331 second resource added, and some differences between the two prey in their relative 332 utilization rates of those resources. This involves simply adding a second resource with 333 logistic growth, characterized by attack rates by the two prey species that differ in their 334 ratio from the two attack rates on resource 1. The simplest situation involves attack rates 335 of prey 1 on the two resources (C11 and C12) that are mirror images of those of the prey 2 336 (C21 and C22); i.e., C11 = C22 and C12 = C21. This set of consumption rates results in 337 competition coefficients that are equal to each other, provided both resources remain in 338 the system. Such a system can be approximated by equations (8), with competition 339 coefficients 12 = 21 = < 1. The equilibrium predator population is plotted against its 340 mortality rate for different values of in figure 4. As is decreased from unity, the 341 predator mortality rate required to exclude the less vulnerable prey 2 increases. A 342 decrease in also decreases the range of mortality rates over which predator population 343 size increases with increasing mortality. Thus, predator population size is often relatively 344 insensitive to its own mortality over a wide range of mortality rates, as shown in the 345 bottom panel of fig. 4. When the competition coefficients are small enough ( < K2 = 346 0.5, given the other parameters in fig. 4), the equilibrium predator population decreases 347 monotonically with its own mortality rate. Of course, if one or both prey were capable of 348 adaptive adjustment of a trait related to both intrinsic growth rate and predator 349 vulnerability, the predator could decrease with harvest even when there was no 350 competition between the prey; this case would be similar to the first model analyzed here. 17 351 352 Discussion 353 We will refer to the increase in population size that is caused by higher mortality 354 as the ‘hydra effect’, in honour of the mythological creature that grew two new heads for 355 every one that was removed. The implications for fisheries policy of the hydra effect are 356 quite profound. Such increases are likely to be interpreted as a license to increase 357 harvesting rates further. However, the increase in the predator in these situations also 358 entails a decrease in the remaining capacity of the prey to increase in population size 359 and/or vulnerability. There is also a decrease in the capacity of the predator to assimilate 360 more prey because its functional response approaches its saturation level. Thus, at some 361 point, the predator population will start to decrease rapidly in response to further 362 increases in harvest, as in figure 1. Of course, the equilibrium population size (shown in 363 figs. 1, 3 and 4) is not attained immediately; this time lag is illustrated in figure 2. In 364 addition, if mortality of the predator increases continuously over time, the population lags 365 behind its equilibrium value, and the final decrease in the population does not have as 366 steep a slope as these graphs of equilibrium densities might suggest. This is shown in 367 Abrams (2002), which compares the change in actual predator population and 368 equilibrium for several models of cycling predator-prey systems in which neither species 369 has adaptive change. That article also shows the population size of a predator may be a 370 large fraction of its original equilibrium size at the point where a steadily increasing 371 mortality passes the level where its equilibrium population size is zero. This means that 372 there is a risk that restrictions on harvest will come too late to prevent a catastrophic 18 373 decline or extinction. This same message applies to the models with adaptive change 374 discussed here. 375 Even if a stock never exhibits the hydra effect (an increase in population size with 376 harvesting), it is likely that both adaptive changes in its characteristics and its interactions 377 with other species will affect population dynamics. Thus, both factors may often need to 378 be studied to predict responses to harvesting or to environmental changes. There have 379 been other calls for more consideration of behavioural effects in understanding fisheries 380 systems (e.g. Dill et al. 2003), but we have previously had few examples of when and 381 why adaptive behaviour might alter the predictions of single-species approaches. 382 The predator's functional response has been shown to play an important role in the 383 results obtained here. If the predator's response is linear (Holling type 1), neither of the 384 categories of model studied here (adaptive prey or replacement of competing prey) 385 predicts a hydra effect in the predator population. This is also true of at least some other 386 functional responses that lack satiation. Here we briefly describe some unpublished 387 results on the effects of changing the 2-prey model (eqs (6)) so that it has the simplest 388 predator functional responses used in the popular computer package, Ecosim (Walters et 389 al. 1997, 2000). In this case, the functional response of a predator with population P to 390 prey species i, with population Ni is assumed to be, CiNi/(Qi + wiP), where Ci, Qi, and wi 391 are constants. This is based on a model of random movement between vulnerable and 392 protected states (Abrams and Walters 1996), rather than the adaptive defense assumed by 393 eq (3) in our analysis. Because of the lack of predator satiation in this response, the 394 effect of mortality on the predator is similar to that in a comparable model with linear 395 functional responses; i.e., no change in equilibrium predator density. We could also 19 396 modify eqs (6) so that the two prey species as well as the predator, have the simplest 397 possible Ecosim functional responses. Because of the dependence of prey functional 398 responses on their own population density, competition between the prey is reduced and 399 the model becomes more similar to eqs (8). As in that model, when there is no satiation, 400 the equilibrium predator density always declines with increases in mortality. Ecosim 401 includes options for including both variable foraging and handling time. These use an 402 iterative numerical procedure to determine handling and foraging times (Walters and 403 Martell 2004). While the mathematical form of the resulting functional response differs 404 from the responses used here, it shares the property of approaching saturation at high 405 population levels. Any such response will often lead to predator populations that increase 406 with increasing harvest rates, given parameter values and food web configurations similar 407 to those explored here. There is at least some evidence for strong satiation in some fish 408 species (e.g. Eby et al. 1997), and Koen-Alonzo and Yodzis (unpub. results) have found 409 at least one instance of stock size increasing with harvesting in an ecosystem model with 410 saturating functional responses. 411 The models analyzed here have all been quite simple, and it is important to 412 determine to what extent our conclusions depend on that simplicity. One question is 413 whether competitive interactions with other predators can prevent the kind of increase in 414 abundance with increased mortality that is demonstrated here. An increase in a second 415 predator following increased mortality of the first might prevent any increase in prey 416 abundance or vulnerability from increasing the first predator. Our models have also 417 assumed that the populations are unstructured; all individuals within a species are 418 identical with respect to both their interactions with other species and their susceptibility 20 419 to harvest. There are many circumstances when structured population models have 420 qualitatively different dynamics than do unstructured population models (e.g., Iwasa et al. 421 1987). Finally, it is important to investigate how coupled adaptive changes in both 422 predator and prey are likely to affect the responses of the populations of either or both 423 species to exploitation. It will require additional work to determine whether models 424 incorporating these additional features differ systematically in the likelihood that a 425 harvested species will increase over a significant range of harvest rates. 426 Although there are many examples of fisheries that have been overexploited, we 427 do not know of any for which there is clear evidence that increased fishing pressure 428 caused an increase in the population size of top predator species, or conversely, that 429 decreased fishing pressure resulted in a decrease in stock size. We have searched R.A.M. 430 Myers’ online database (http://ram.biology.dal.ca/~myers/data.html), which provides 431 estimates of both stock size and fishing mortality for a number of species. There are 432 some examples where decreases in stock size occurred simultaneously with a decrease in 433 fishing mortality (e.g., monkfish, Lophius piscatorius, data from the Report of the ICES 434 Working Group on the Assessment of Southern Shelf Demersal Stocks). However, this 435 pattern is also consistent with a negative effect of harvesting on stock size, provided the 436 decrease in fishing mortality is not sufficiently large. For most top predators in Myers’ 437 database, stock estimates are unavailable until after fishing pressure had been intense for 438 some time. Thus, in most of the data sets having estimates of fishing mortality, the 439 mortality was already relatively high when it was first measured (or at least reported). 440 As far as we have been able to determine, past records of harvest and stock size do not 441 provide clear evidence for the phenomena we have predicted here. 21 442 It is possible that increases in stock size due to harvesting has never occurred 443 because of some complexities of natural communities that are missing from the models 444 considered here. It is also possible that such increases occurred, but were not observed 445 because reliable information on stock size is often not available during the early stages of 446 a fishery on a given species. Because of environmental variability and variation in 447 fishing rates, it may be difficult to assign a cause to an observed increase in abundance. 448 It is also possible that the high initial rate and rapid increase in harvest rates for many fish 449 stocks have meant that the fishery collapsed before any of the compensatory processes 450 discussed here had time to operate. Restrepo et al. (1998) argue that the natural mortality 451 rate may be used as a reasonable proxy for the MSY fishing mortality rate. Harvest rates 452 in modern fisheries are usually much larger than the natural mortality rate, and they have 453 increased rapidly when stock sizes have remained reasonably high (Pauly et al. 2002). In 454 addition, modern fisheries have often involved harvesting of several trophic levels, or 455 sequential harvest of different predatory species (Jackson et al. 2001). Any of these 456 conditions may have prevented an increase in predator stock sizes in response to changes 457 in the population densities and/or defensive traits of their prey. 458 Recent documentation of extensive declines in predatory fish species have 459 brought about calls for greater consideration of the maintenance of biological diversity in 460 fisheries regulation (e.g. Pauly et al. 2002). Such a shift would likely require that future 461 fisheries not increase harvest rates as rapidly as has been typical in the past. This would 462 make it possible to determine whether the ‘hydra effects’ described here and in Matsuda 463 and Abrams (2004) are likely to occur in natural fish communities. 464 465 22 466 Acknowledgements 467 The work was supported by a Strategic Project grant from the Natural Sciences and 468 Engineering Research Council of Canada to P.A. and a grant from the Japan Society for 469 Promotion of Science to H. M. We thank C. J. Walters and D. L. DeAngelis for their 470 comments on an earlier draft. 471 23 472 References 473 Abrams, P. A. 1999. Is predator mediated coexistence possible in unstable systems? 474 475 Ecology 80: 608-621. Abrams, P. A. 2001. Modeling the adaptive dynamics of traits involved in inter- and 476 intra-specific competition: An assessment of three methods. Ecol. Lett. 4: 166- 477 175. 478 479 480 481 482 Abrams, P. A. 2002. Will small population sizes warn us of impending extinctions? Am. Nat. 160: 293-305. Abrams, P. A., and Matsuda, H. 1997. Prey evolution as a cause of predator-prey cycles. Evolution 51: 1740-1748. Abrams, P. A., Matsuda, H., and Harada, Y. 1993. Evolutionarily unstable fitness 483 maxima and stable fitness minima in the evolution of continuous traits. Evol. 484 Ecol. 7: 465-487. 485 Abrams, P. A., Namba, T., Mimura, M., and Roth, J. D. 1997. Comment on Abrams and 486 Roth: The relationship between productivity and population densities in cycling 487 predator-prey systems. Evol. Ecol. 11: 371-373. 488 489 490 491 492 493 Abrams, P. A. and Walters, C. J. 1996. Invulnerable prey and the statics and dynamics of predator-prey interactions. Ecology 77: 1125-1133. Armstrong, R. A. 1979. Prey species replacement along a gradient of nutrient enrichment: a graphical approach. Ecology 60: 76-84. Dieckmann, U. and Law, R. 1996. The dynamical theory of coevolution: a derivation from stochastic ecological processes. J. Math. Biol. 34: 579-612. 24 494 Dill, L. M., Heithaus, M. R., and Walters, C. J. 2003. Behaviorally mediated indirect 495 interactions in marine communities and their conservation implications. Ecology 496 84: 1151-1157. 497 Eby, L. A., Rudstam, L. G. and Kitchell, J. F. 1995. Predator responses to prey 498 population dynamics-an empirical analysis based on lake trout growth rates. Can. 499 J. Fish. Aquat. Sci. 52: 1564-1571. 500 Eklöv, P. and Persson, L. 1995. Species-specific antipredator capacities and prey refuges: 501 Interactions between piscivorous perch (Perca fluviatilis) and juvenile perch and 502 roach (Rutilus rutilus) Behav. Ecol. Sociobiol. 37: 169-178. 503 504 505 Fraser, D. F. and Gilliam, J. F. 1992. Nonlethal impacts of predator invasion: Facultative suppression of growth and reproduction. Ecology 73: 959-970. Grover, J. P. and Holt, R. D. 1998. Disentangling resource and apparent competition: 506 Realistic models for plant-herbivore communities. J. Theor. Biol. 191: 353-376. 507 Heino, M. and Godø, O. R. 2002. Fisheries-induced selection pressures in the context of 508 509 510 511 sustainable fisheries. Bull. Mar. Sci. 70:639-656. Hilborn, R. and Walters, C. J. 1992. Quantitative Fisheries Stock Assessment: Choice, Dynamics and Uncertainty. Chapman and Hall, NY. Holling, C. S. 1959. The components of predation as revealed by a study of small 512 mammal predation of the European pine sawfly. Can. Ent. 91: 293-320. 513 Hollowed, A. B., Bax, N., Beamish, R., Collie, J., Fogarty, M., Livingston, P., Pope, J., 514 and Rice, J. C. 2000. Are multispecies models an improvement on single-species 515 models for measuring fishing impacts on marine ecosystems? ICES J. Mar. Sci. 516 57: 707-719. 25 517 518 519 520 521 Holt, R. D., Grover, J. P., and Tilman, D. 1994. Simple rules for interspecific dominance in systems with exploitative and apparent competition. Am. Nat. 144: 741-771. Iwasa, Y., V. Andreasen, and Levin, S. A. 1987. Aggregation in model ecosystems. I. Perfect aggregation. Ecol. Model. 37: 287-302. Jackson, J. B. C., Kirby, M. X., Berger, W. H., Bjorndal, K. A., Botsford, L. W., 522 Bourque, B. J., Bradbury, R. H., Cooke, R., Erlandson, J., Estes, J. A., Hughes, T. 523 P., Kidwell, S., Lange, C. B., Lenihan, H. S., Pandolfi, J. M., Peterson, C. H., 524 Steneck, R. S., Tegner, M. J., and Warner, R. R. 2001. Historical overfishing and 525 the recent collapse of coastal ecosystems. Science 293: 629-638. 526 Latour, R. J., Brush, M. J., and Bonzek, C. F. 2003. Towards ecosystem-based fisheries 527 management: Strategies for multispecies modeling and associated data 528 requirements. Fisheries. 28(9): 10-22. 529 Leibold, M. A. 1996. A graphical model of keystone predators in food webs: trophic 530 regulation of abundance, incidence and diversity patterns in communities. Am. 531 Nat. 147: 784-812. 532 Lima, S. L. 1998. Stress and decision making under the risk of predation: Recent 533 developments from behavioral, reproductive, and ecological perspectives. Adv. 534 St. Behav. 27: 215-290. 535 536 537 Matsuda, H. and Abrams, P. A. 2004. Effects of predator-prey interactions and adaptive change on sustainable yield. Can. J. Fish. Aquat. Sci. 61:175-184. Pauly, D., Christensen, V., Guenette, S., Pitcher, T. J., Sumaila, U. R., Walters, C. J., 538 Watson, R., and Zeller, D. 2002. Towards sustainability in world fisheries. Nature 539 418: 689-695. 26 540 541 542 Quinn II, T.J., and DeRiso, R.B. 1999. Quantitative Fish Dynamics, Oxford University Press, NY. Restrepo V. R., Thompson, G. G., Mace, P. M., Gabriel, W. L., Low, L. L., MacCall, A. 543 D., Methot, R. D., Powers, J. E., Taylor, B. L., Wade, P. R., Witzig, J. F. (1998) 544 Technical Guidance On the Use of Precautionary Approaches to Implementing 545 National Standard 1 of the Magnuson-Stevens Fishery Conservation and 546 Management Act. NOAA Tech. Memo. NMFS-F/SPO-39, 54pp. 547 548 549 550 551 552 553 Reznick, D. N. Bryga, H., and Endler, J. A. 1990. Experimentally induced life-history evolution in a natural population. Nature 346: 357-359. Rosenzweig, M. L. and MacArthur R. H. 1963. Graphical representation and stability conditions for predator-prey interactions. Am. Nat. 97: 209-223. Taylor, P. D. and Day, T. 1997. Evolutionary stability under the replicator and the gradient dynamics. Evol. Ecol. 11: 579-590. Turner, A. M. and Mittelbach, G. G. 1990. Predator avoidance and community structure: 554 Interactions among piscivores, planktivores, and plankton. Ecology 71: 2241- 555 2254. 556 Walters, C. J., Christensen, V., and Pauly, D. 1997. Structuring dynamic models of 557 exploited ecosystems from trophic mass-balance assessments. Rev. Fish. Biol. 558 Fish. 7: 1-34. 559 560 Walters, C. J. and S. Martell. 2004. Fisheries ecology and management. Princeton Univ. Press. Princeton, N. J. 561 Walters, C. J., Pauly, D., Christensen, V., and Kitchell, J. F. 2000. Representing density 562 dependent consequences of life history strategies in aquatic ecosystems: EcoSim 27 563 564 565 II. Ecosystems 3: 70-83. 28 566 567 Appendix Stability and responses of equilibrium values to predator mortality in Eqs (1a, 1b, 3) To determine the change in the equilibrium R, N, and C with an increase in d, we 568 569 set the left hand sides of equations (1a, 1b, 3) equal to zero, and differentiate the resulting 570 equations with respect to d, noting that the equilibrium values of the variables are implicit 571 functions of d. This yields 3 simultaneous linear equations for the three quantities, 572 N*/d, C*/d, and R*/d. As an example, differentiating equation (1a) at 573 equilibrium yields the following equation: 574 0 f ' 575 Solving this, together with the corresponding equations derived from the equilibrium 576 conditions for equations (1b) and (3), yields expressions showing how the equilibrium 577 values of the three variables change with the predator's death rate, d: 578 2 2 f f 2 f f 2 f Ng ' C R C 2 C 2 C R N * C R R d 2 2 f f 2 f gb ' g CRg ' C R C 2 C R R C (A1) 579 f 2 f C R CR C * d 2 2 f f 2 f b ' g CRg ' C R C 2 CR R C (A2) 580 2 f C CR R * d 2 2 f f 2 f b ' g CRg ' C R C 2 CR R C (A3) R * R * N * C * C * CNg ' R C Ng Cg d d d d d 29 581 The conditions for the local stability of the equilibrium point(s) are found from the 582 Jacobian matrix of the dynamic system eqs (1a, 1b, 3) evaluated at the equilibrium point. 583 Unfortunately, the full conditions for stability (conditions that the eigenvalues of this 584 matrix have negative real parts) are very complex. However, one of the necessary 585 conditions is that the quantity in square brackets in the denominators of the above three 586 expressions must be negative. (This quantity is equal to a positive constant multiplied by 587 the determinant of the Jacobian matrix.) 30 588 Figure Legends 589 590 Figure 1. The equilibrium predator population size as a function of its own per capita 591 mortality rate for a system in which dynamics are described by equations (1a, 1b, and 3), 592 using the functional forms given by eqs (5). The model also included minimum and 593 maximum values of 0 and 5 respectively for the trait C. The trait was prevented from 594 exceeding these bounds by adding a function, /C2 - /(5 – C)2, to the right hand side of 595 eq (3), with = 10-6. This provides a force that pushes C away from these boundaries, 596 but has a negligible effect when C is significantly different from 0 or 5. The parameter 597 values from eqs (5) are: rm = 2; r1 = 2; r2 = 0.2; K = 1; h = 0.5; b0 = 1. If adaptation is 598 slow (approximately v < 0.01), the equilibrium is stable over most the entire range of 599 mortality rates shown (approximately d > 0.15). More rapid adaptation produces limit 600 cycles over a broader range of low mortality rates. For example, if v = 0.5 the system 601 always exhibits cycles for d < 1.37, and there are alternative cyclic and noncyclic 602 attractors for approximately 1.37 < d < 1.68. 603 604 Figure 2. The transient responses of prey, predator and trait, to a doubling of the predator 605 mortality at time 0, for the system discussed in figure 1. The rate constant for change in 606 C was v = 0.05. The initial system was at equilibrium for the parameter values given in 607 fig. 1 with a predator mortality rate of d = 0.75. This mortality was doubled to d = 1.5 at 608 time 0. 609 31 610 Figure 3. The equilibrium abundance of the predator as a function of its own death rate. 611 The underlying model eqs (8), in which the resource dynamics are not explicit and the 612 prey compete via Lotka-Volterra competition. The parameters are, 12 = 21 = 1; K1 = r1 613 = r2 = 1; h = 1; s1 = 1; B0 = 1. Each line shows the relationship for the value of K2 given 614 above the line. 615 616 Figure 4. The equilibrium abundance of the predator as a function of its own death rate, 617 for a range of different strengths of competition. The model is given by equations (8) and 618 it is assumed that the two competition coefficients, ij, are both 0.5. The other parameter 619 values are the same as in figure 3.