When does greater mortality increase population size? The long history and diverse mechanisms underlying the hydra effect

Authors


*E-mail: peter.abrams@utoronto.ca

Abstract

The phenomenon of a population increasing in response to an increase in its per-capita mortality rate has recently been termed the ‘hydra effect’. This article reviews and unifies previous work on this phenomenon. Some discrete models of density-dependent growth were shown to exhibit hydra effects in 1954, but the topic was then ignored for decades. Here the history of research on the hydra effect is reviewed, and the key factors producing it are explored. Mortality that precedes overcompensatory density dependence always has the potential to produce hydra effects. Even when mortality follows density dependence, hydra effects may occur in unstable systems due to changes in the amplitude and/or form of population cycles. An increase in resource productivity due to lower consumption rates following increased consumer mortality can also produce a hydra effect. Lower consumption can come about as the result of increased satiation of the consumers or changes in behaviour of either consumer or resource species that reduce the mean attack rate. Changes in species composition of a resource community may also decrease the average attack rate. Population structure can promote hydra effects by allowing separation of the timing of density dependence and mortality, although stage-specific density dependence usually decreases hydra effects.

Introduction

It is generally assumed that increased mortality applied to a population will decrease the size of that population. This logic underlies many strategies in fisheries and pest management and in conservation biology. The traditional connection between mortality and population size also affects the definitions of interspecific interactions in a fundamental way. The most common definition of an interaction (Abrams 1987) is based on the change in the density of a target species given a sustained change in the density of a manipulated species (a press perturbation sensuBender et al. 1984). However, press perturbations of population sizes are extremely difficult to achieve, as they require continual monitoring of population size with continuous adjustment by culling or introductions. On the other hand, it is often relatively easy to manipulate absolute or per-capita mortality rates. Using altered mortality as a surrogate for a press perturbation of population size would not be a problem if higher mortality applied to a species always implies a lower population size. Unfortunately, this is not the case. Thus, it is important to determine under what circumstances and how often population sizes increase with increased mortality in natural communities. This phenomenon will be referred to here as the ‘hydra effect’ (following Abrams & Matsuda 2005), after the mythological beast that grew two heads to replace each one that was removed.

In spite of the fact that many commonly used models often predict hydra effects, evidence for their existence in natural communities is scarce (see ‘Relevant empirical evidence’ in the Discussion section below). This may reflect a lack of appropriate observations; if not, the scarcity calls for an explanation based on some lack of correspondence between the structure of natural communities and the structure of commonly used models. There is little doubt that the lack of empirical evidence is in part because previous theoretical work on this concept has frequently been forgotten or misinterpreted (see Discussion). In addition, models that have produced hydra effects have had a variety of forms, assumptions and methods of analysis. It has not been clear what features or assumptions are necessary or sufficient for producing such effects. I argue here that three factors are responsible for hydra effects: altered patterns of population fluctuations; temporal separation of mortality and density dependence; and reductions in resource exploitation rates. The article begins with a brief review of the history of work on this topic and ends with a discussion of potential empirical examples. Several new theoretical results are derived to generalize previous findings.

History

The possibility of increased population size under increased mortality seems to have first been recognized by Ricker (1954), although this aspect of his often cited article has largely been ignored. He derived this result for the discrete model of density-dependent population growth that now bears his name, and he assumed that mortality preceded density dependence. Ricker (1954, p. 582) stated, ‘Moderate destruction of adults will in general tend to stabilize the population at or about some magnitude greater than its primitive average abundance’. He argued that this was likely to be of practical importance in pest management. Ricker’s (1954) study remained as the only explicit discussion of population size increasing in response to mortality until the late 1990s. Early theory about predator–prey interactions by Rosenzweig & MacArthur (1963) uncovered a mechanism by which mortality applied to a predator population could potentially increase its equilibrium population size. Their graphical analysis contained the implicit result that equilibrium predator population size could increase with predator mortality when the equilibrium was unstable. However, Rosenzweig and MacArthur did not mention this possibility in their article (contrary to what is claimed by some recent references to that work; e.g. De Roos et al. 2007). In addition, the equilibrium is unstable for all cases where mortality increases the equilibrium predator population in their model (see below), and the average predator population size need not change in the same direction as the equilibrium in response to a perturbation such as increased mortality. Subsequent numerical work on a variety of other models of interacting species with sustained fluctuations has revealed that average and equilibrium population sizes often respond in different directions to a given perturbation of a cycling population (Abrams & Roth 1994; Abrams et al. 1997, 2003; Brassil 2006).

The theoretical possibility of increased population size with increased mortality again began to receive some attention from theorists starting roughly a decade ago. Works that mention such an outcome include: Jonzén & Lundberg (1999), Boyce et al. (1999), Abrams (2002, 2005), Abrams & Vos (2003), Abrams et al. (2003), Matsuda & Abrams (2004), Abrams & Matsuda (2005), Abrams & Quince (2005), Ranta et al. (2006), De Roos et al. (2007), Matsuoka & Seno (2008), Ratikainen et al. (2008), Schreiber & Rudolf (2008) and Seno (2008). This growing list makes it important to analyse both the differences and commonalities of the scenarios employed in different studies. Most of these studies have assumed highly specified models, so that the generality of the conditions under which hydra effects might occur were not evident.

Mechanisms underlying the hydra effect

For some large classes of models, increased mortality must cause decreased population size. This is true, for example, of any continuous unstructured model of density-dependent population growth in a constant environment. Such models have the following general form:

image(1)

where N is population size and f is the density-dependent per-capita growth rate (including natural mortality). Finally, d is an additional density-independent mortality rate, which could be due to adverse environmental conditions or a constant harvesting effort. See Appendix S1A, in Supporting Information for a short derivation of the result that the equilibrium density, Neq, decreases with d. This result and the popularity of models (like the logistic) that have the form of eqn  1 have contributed to the idea that mortality always decreases the population size. However, eqn  1 has a number of features that are far from universally applicable; it does not permit sustained cycles; it necessarily implies simultaneous operation of density dependence and density-independent mortality; and it does not allow for mortality to alter the form of the growth function f. One or more of these features has been absent in previous models that exhibited hydra effects. The corresponding mechanisms producing hydra effects are treated in turn below. The different models can be classified by whether they are discrete or continuous, whether the population modelled is homogeneous or structured (e.g. by stage or age), and by whether the model does or does not explicitly include interacting species.

Mechanism 1: altered variation in population size

This mechanism assumes that the populations have sustained fluctuations. In the scenarios considered here, these fluctuations are generated endogenously, by time lags or interactions between species. The essential mechanism is that fluctuations usually alter the temporal mean density of a population (Armstrong & McGehee, 1980), and mortality usually alters the nature of population fluctuations. Discrete analogues of eqn  1, which necessarily incorporate a time lag, are considered first.

Models with discrete generations and density-dependent population growth

Discrete time models of unstructured populations can exhibit sustained nonequilibrium dynamics, as noted by Ricker (1954) and many subsequent authors. This possibility arises because the time-lag inherent in discrete time models allows the population to overshoot its equilibrium by more than its original displacement from the equilibrium (Otto & Day 2007). The resulting population fluctuations can change average densities, and it is well known that the pattern of fluctuations in such models changes with the maximum per-capita growth rate. Maximum growth decreases as harvest increases. Consider the discrete analogue of the continuous eqn  1, where Nt is now the population size at the beginning of generation t:

image( (2a))

Here f is a decreasing function of Nt giving the proportional change in the population size in one time unit. A commonly used example of eqn 2a is the discrete logistic:

image( (2b))

where f(N) = r(1 − N). Here N is measured in units of its maximum density; N cannot exceed 1. The equilibrium population size is Neq = 1 − 1/r. The dynamics of this model are covered in many textbooks (e.g. Hastings 1997; Otto & Day 2007); a large enough r leads to regular or chaotic cycles and r > 4 implies extinction because it allows the possibility of a crash to negative population sizes. Mortality that follows the action of density dependence multiplies the right-hand side of eqn  2b by the survival fraction (1 − d). As a result, increasing mortality is simply equivalent to reducing r; the equilibrium density becomes Neq = 1 − 1/(r(1 − d)), which clearly decreases with increasing d. However, when the population cycles [r(1 − d) > 3], the equilibrium is unstable, so this density is never observed.

Equation 2b is probably the best-known single-species model of density-dependent growth in existence, so it is surprising that the impact of cycles on average population size in unstable systems has only recently been illustrated. Seno (2008, p. 68) showed that when r is initially close to its maximum value of 4, increasing d changed the average population size in an irregular, but generally increasing manner until the equilibrium became stable. The increasing trend results from a similar trend towards less-extreme fluctuations in the population size. A higher mortality can reduce the occurrence of extreme population crashes that require many generations for population recovery, and this increases the average population size. However, this is not the only impact of greater mortality within the unstable region. Particular types of dynamics are associated with different average densities. For example, three-point cycles have low mean densities because these cycles consist of a slow 2-year increase from low levels followed by a single large decrease. Easily obtained analytical results (Abrams, P.A., unpublished data) show that the mean population size decreases with d when the dynamics are four-point cycles, but increases when r is low enough that there are two-point cycles. The net result is that, when the initial r is close to its maximum of r = 4, the trajectory of the mean population size vs. mortality is very irregular (see Seno 2008), in spite of a general upward trend when d is relatively small.

It is useful to explore the impact of mortality that acts before density dependence on cycles and average densities using a more flexible and biologically realistic model (i.e. one that cannot predict negative densities). The ‘Maynard Smith’ or ‘extended Beverton–Holt’ model (Maynard Smith 1974) is relatively flexible, and unlike the discrete logistic, it does not predict deterministic extinction above any value of the maximum rate of increase. Here the population dynamics are given by:

image(3)

where the positive exponent, m, and the constant α determine how density affects the per-capita reproductive rate, and B is the maximum per-capita rate of increase. This model reduces to a Beverton & Holt’s (1957) relationship when m = 1. Exponents m > 1 produce overcompensatory density dependence, because an increase in Nt may decrease Nt+1. Because stability is not affected by α, this parameter is set to 1 in the numerical examples explored here. Mortality that acts after density dependence is again incorporated by multiplying Nt in the numerator of eqn  3 by (1 − d). Figure 1 shows the impact of such mortality on the mean population size for two cases with strong density dependence and high enough maximum rates of increase to cause sustained fluctuations. Both the equilibrium and mean population sizes are shown; these are identical when the system is stable. In Fig. 1a (m = 30; B = 4), the ‘shape’ parameter m is very high, implying that large population sizes can produce crashes to very low densities. In this case, the mean N is much smaller than the equilibrium for most cycling systems because of the long time periods spent recovering from population crashes. The mean increases sharply when mortality levels are sufficient to greatly increase the minimum population size, and it is maximized where the system becomes stable. In Fig. 1b, m = 6 and B = 20. Here, there is an abrupt, but small increase in the mean population size near d = 0.25, associated with a transition from chaotic dynamics to a period-5 cycle. There is also a slight increase between d = 0.9 and d = 0.91, associated with a transition from chaos to regular cycles. However, the clear trend is for smaller average population sizes as mortality increases, as predicted by the equilibrium population size. It should be noted that, when significant temporal variation in B is incorporated into the Fig. 1b example, these irregularities in the Neq vs. d relationship do not occur, and Neq decreases nearly monotonically with d. In both cases illustrated, the mean population size drops very abruptly as d approaches the maximum value allowing persistence. Ricker’s (1954) model, investigated by Seno (2008), also displays a decline in the mean population size when mortality is applied after density dependence, even when the population exhibits chaotic dynamics (Seno 2008; Fig. 2c).

Figure 1.

 Population size vs. mortality for examples of the Maynard Smith model (eqn  3) when mortality acts after density dependence. The parameters of the two examples are: (a) B = 4, α = 1 and m = 30, and (b) B = 20, α = 1, and m = 6. The dashed line gives the equilibrium population, while the solid line gives the average population. These differ when the equilibrium is unstable.

Figure 2.

 Predator population size (N) as a function of per-capita mortality, d, in a simple predator–prey system with a type-2 functional response and θ-logistic prey (R) growth. The model is given by, inline image where r and K are prey intrinsic growth rate and carrying capacity, I is prey immigration and θ is the exponent of density dependence. B, C, h and d are respectively: conversion efficiency, attack rate, handling time and per-capita mortality. Parameter values common to (a) and (b) are {r = K = h = 1; C = 5; B = 0.1; θ = 1}. In (a), there is no immigration of prey from outside the system, while in (b), immigration is I = 0.0001. Figure (c) and (d) are modifications of the system with parameters {r = K = h = 1; C = 2.5; B = 0.5; θ = 2}; figure (c) again assumes I = 0, while (d) assumes I = 0.0001. Small irregularities in the lines giving average population sizes are the result of averaging over a fixed period that is a non-integer multiple of the cycle period. In all panels, the dashed line gives inline image assuming I = 0.

The conclusion from the above is that, while mortality may increase the population size as the result of changing cycle amplitude, this requires that the original cycles be characterized by rather long periods at low densities. There can be quite large increases in the mean density with small increases in mortality under these conditions. Unfortunately, these simple discrete models of density dependence are difficult to relate to population interactions, and they lack population structure and delayed effects longer than a single time unit. In general, density dependence arises from some form of resource limitation, so it is important to ask whether discrete or continuous consumer–resource models display hydra effects when the shape of population cycles changes with mortality.

Consumer–resource models

In most widely used models that can exhibit sustained cycles, mortality affects whether cycles occur as well as cycle amplitude (Rosenzweig 1971; Gilpin 1975; Hastings 1997). Here, mortality again has rather complicated effects on the mean population size of the species. The impacts of mortality on the mean consumer density in simple predator–prey models with cycles have been studied before, but only for a few numerical examples (Abrams 2002; Matsuda & Abrams 2004). This question is re-examined for a wider range of such models here.

I start by reviewing work on simple predator–prey models having unstructured populations, logistic prey growth and a Holling type-2 predator functional response (Holling 1959). The effect of mortality on equilibrium and average population sizes in such models was treated in studies by Abrams (2002) and Matsuda & Abrams (2004). The equilibrium population size of the predator increases with mortality over the entire range of mortalities where the equilibrium point is unstable; this increase can be quite substantial when handling time is large, as shown in Abrams (2002; his Fig. 1, p. 295). The maximum equilibrium population size occurs at the mortality rate where the system is on the border between stability and cycles. The reason for this increase in equilibrium is actually the third mechanism (see below); high mortality indirectly decreases the predator’s capture rate by increasing the saturation of the type-2 functional response. The resulting decrease in overexploitation means that higher prey intake more than offsets the increased mortality, and the equilibrium predator density increases. The same outcome can result from an Allee effect in the prey growth, which again implies overexploitation at low mortalities. However, this equilibrium is never observed in simple predator–prey models with prey-dependent functional responses because the equilibrium is unstable. This again requires the calculation of mean densities.

The nonlinear functional response means that, when mortality alters the nature of population fluctuations, it also affects the temporal average density. In most cases, the mean consumer density is reduced relative to equilibrium in fluctuating predator–prey systems, and is reduced more, the greater the amplitude of the fluctuations. This is a result of the saturating functional response, which prevents the predator from being able to make full use of high resource densities. Because the amplitude of cycles usually decreases as mortality is increased and the system approaches stability, this change in cycle amplitude produces a steep increase in the mean density over a range of mortalities close to the mortality that stabilizes the system (Matsuda & Abrams 2004). However, several features of consumer–resource systems can reverse the effect of mortality on cycle amplitude, over at least some ranges of mortality rates. These features include type-3 consumer responses and resource immigration.

Figure 2 provides examples of the potential relationships between consumer mortality and average consumer density in a slight generalization of the logistic/type-2 response model discussed above. The resource growth function is generalized to the theta-logistic (Gilpin and Ayala 1973), and immigration of resource from outside the system is possible, resulting in the model given in the legend of Fig. 2. All of the panels in Fig. 2 describe systems that are unstable for a wide range of mortality rates. In Fig. 2a,b, the impacts of a low level of immigration are illustrated for pure logistic growth. In the complete absence of resource immigration (Fig. 2a), there is a positive effect of mortality on average population size in cycling systems, although the mean density is considerably less than the equilibrium, and is very insensitive to mortality across most of the range of potential mortality rates. The very small migration rate added in Fig. 2b changes the sign of the response of density to mortality over most of the range of mortalities producing cycles. Here, immigration prevents cycle amplitude from becoming extremely large at very low mortality rates. This allows the consumer to attain relatively high densities at very low mortalities; cycle amplitude increases and Neq decreases with d until both trends reverse at moderate mortality rates. A somewhat different pattern is shown in Fig. 2c,d, which also compare systems that lack (Fig. 2c) or have (Fig. 2d) a low level of immigration. This case is characterized by a lower capture rate than in Fig. 2a,b, which decreases the range of parameters producing instability and reduces cycle amplitudes. The system in Fig. 2c,d also has a slightly different form of density dependence; θ-logistic growth with θ = 2. In the absence of immigration (Fig. 2c), the response is qualitatively similar to that in Fig. 2a, with a slight increase in the mean density with mortality over most mortality rates yielding instability and a very rapid increase near the point where the system stabilizes. With a low rate of immigration (Fig. 2d), the relationship of mean density to mortality is bimodal, with two increasing and two decreasing segments. At very low consumer mortality, most resource reproduction is suppressed because the consumer can gain a large fraction of its needs from the resources coming in from outside the system. As a result resources never achieve high density and consequently have low productivity; for example, at d = 0.01 in Fig. 2d, the maximum resource density during a cycle is only 2.55% of the carrying capacity. Increasing mortality above this value allows more pronounced cycles, and consequently much greater productivity; at d = 0.05 the maximum resource density is over 66.46% of the carrying capacity. For mortalities between c. 0.05 and 0.15 the mean consumer population drops with mortality, as cycle amplitude grows. However, cycles eventually decrease in amplitude as the system approaches the mortality producing stability (c. d = 0.27), resulting in another increase in mean consumer density. Amplitude is not the only factor that affects mean consumer size; higher mortality also reduces the cycle period and decreases the duration of phases with very low resource densities.

The impact of mortality in systems having more than two species is even more complicated. However, it remains true that increased mortality of one species usually alters the nature of endogenously generated cycles, and this change is likely to affect the average population densities of all species. Continuous models of three-level food chains (Hastings & Powell 1991) and systems with four or more species (e.g. Vandermeer 2004) are capable of a wider range of dynamics than the continuous-time two-species models discussed here. The same is true of discrete-time systems having two or more interacting species. There are very few published studies where average densities have been related to mortality in any of these or other multispecies models (one case is the study of a 2-predator–1-prey system by Abrams et al. (2003)). As an inducement for others to pursue this question further, Appendix S2 examines the impact of mortality on mean densities for specific parameter values for the Nicholson–Bailey parasitoid–host model modified by host density dependence (Fig. S1), and Hastings & Powell’s (1991) three-species food-chain model (Fig. S2). In the latter model, mortality of the top species often increases the amplitude of cycles in the two lower-level species, eliminating the hydra effect predicted based on Mechanism 3 below.

Conclusions regarding the impact of fluctuations on hydra effects

The literature review and new analyses presented here confirm that the impact of mortality on average densities is affected by the changes that greater mortality brings about in the form and amplitude of cycles. Average density may increase when the equilibrium decreases, and vice versa. Changes in mean density with mortality need not be monotonic or unimodal. Systems having complex dynamics are good candidates for finding hydra effects. However, such effects are difficult to document because they requires long enough time series to measure average densities. This should be possible in some laboratory systems, but is likely to be much more difficult in the field.

Although this paper concentrates on single homogeneous environments, many populations and communities consist of local patches within which interactions occur, and between which there is some immigration. Gilpin (1975) argued that local predator–prey systems undergoing large amplitude limit cycles were likely to go extinct; prey extinction due to low population sizes would be followed by predator starvation. He argued that this could lead to natural selection for characteristics that reduced cycle amplitude. The same argument suggests that mortality applied to consumers in all patches would decrease cycle amplitude and thereby increase minimum densities. The resulting lower probability of local extinction could increase population densities across the landscape (Holt 2002). This type of hydra effect could operate on any system that undergoes large-amplitude fluctuations in population size when mortality is low, but is stabilized by greater mortality.

Mechanism 2: temporal separation of mortality and density dependence

The fact that the simple continuous time models like eqn  1 cannot exhibit a hydra effect is because density dependence continuously and instantaneously adjusts birth and/or death rates in response to any change in density caused by an imposed mortality. The lack of a hydra effect in stable discrete generation models with mortality following density dependence is a consequence of the fact that density dependence cannot counteract such mortality. Ricker (1954) assumed that mortality precedes density dependence. This allows hydra effects in a wide range of discrete time models for stable as well as unstable systems.

Discrete generation models

Here, I continue to assume the general form of the discrete generation model, eqn  2a, which encompasses Ricker’s model, among others. When mortality (with a fraction, d) precedes density dependence, eqn  2a is changed to have the following form:

image(4)

Differentiation of the condition for equilibrium (Nt +1 = Nt) yields the following expression for the change in equilibrium density with the probability of mortality:

image(5)

where f′ = df/dN. Because f′ < 0, Neq may either increase or decrease with d; the formula implies that increases in N with d are most likely when d is very small; decreases are inevitable when d approaches its maximum of d = 1. The equilibrium N increases with mortality at low mortalities if N0 > −(1/f′)|N0, where N0 is the population size in the absence of harvesting. When d = 0, the equilibrium is stable if N0 > −(2/f′)|N0. Thus, if a hydra effect exists, it must occur in some stable systems. However, there are some models for which the equilibrium N never increases with d. Appendix S1B shows that a hydra effect cannot occur unless density dependence is overcompensatory, defined as a decrease in output from a life-history stage with an increase in input. Overcompensation here means that Nf decreases with N for some range of N. Thus, d(Nf)/dN = f + Nf′ < 0, which can be rewritten as the condition for a hydra effect given above (N > −1/f′). In the Beverton & Holt’s (1957) model f = C1/(1 + C2N), where the Ci are constants; here the condition for overcompensation cannot be satisfied (Seno 2008). Expression (5) does not apply to the average densities in systems that cycle. Cycles do occur for the discrete logistic model, and Fig. 5 in Seno (2008) shows that mortality that precedes density dependence can increase the mean population size in this model, even when the equilibrium decreases with d.

If the mortality operates continuously over the density-dependent part of the lifespan with an instantaneous rate, D, the mean population in generation t is the initial value, Nt multiplied by the factor, exp(−D)/D. When density dependence is based on the mean population size over a generation t, the general expression for dynamics becomes:

image(6)

Here there is no separation in time between mortality and density dependence. Appendix S1C analyses the impact of D on Neq. It shows that the equilibrium N must decline with D when there is a stable equilibrium density. Seno (2008) derives some results assuming that mortality acts at a single point in time within the season, and shows that the hydra effect occurs over a wider range of mortality rates, the earlier in the season the mortality is applied.

The flexible Maynard Smith (1974) model provides a useful example to illustrate the magnitudes of hydra effects that occur when mortality precedes density dependence. Some formulas for stability and equilibrium density in this model are provided in Appendix S1D. Figure 3 illustrates the relationship between mortality rate and equilibrium population size for two exponents, m. Figure 3a assumes m = 20/9, which is the largest exponent for which the system is stable at all mortalities for the maximum birth rate assumed (B = 10). Figure 3b illustrates the same relationship for equilibrium and mean population sizes, given an exponent (m = 4) that produces cycles or chaos over a large range of mortalities. Both of these cases (and generally m > 2) produce a strong hydra effect, which operates over the vast majority of the range of potential mortalities when mortality acts before density dependence. In cases with pronounced hydra effects, a small increase in mortality reduces the population from its maximum size to zero.

Figure 3.

 The equilibrium population size as a function of the fraction of individuals harvested (d), for the Maynard–Smith model (eqn 3) when mortality operates before density dependence. In panel (a), parameters are B = 10, α = 1 and m = 20/9. This value of m is as large as it can be and still produce a stable equilibrium across the full range of possible d values. (b) The equilibrium (dashed) and average (solid) population sizes in the Maynard Smith model. Parameters are as in (a) except that m = 4, producing more strongly nonlinear density dependence, which leads to instability for most mortalities.

When organisms use different resources or occupy different habitats during the course of a generation, both mortality factors and density dependence usually differ in the different stages. Jonzén & Lundberg (1999) illustrated a stage-specific hydra effect in a discrete generation model with two sequentially density-dependent life-history stages preceding reproduction. (A hydra effect is stage-specific if the abundance of one stage increases with greater mortality in that stage.) Several subsequent discussions of hydra effects in discrete models have reviewed Jonzén and Lundberg’s results (Boyce et al. 1999; Ranta et al. 2006; Ratikainen et al. 2008), without mentioning that their assumed temporal structuring of density dependence was not required for the hydra effect they described. The fact that strong density dependence in the first period necessarily leads to weaker density dependence in the second (and vice versa) actually makes life histories with two (or more) sequential bouts of density dependence less likely to exhibit hydra effects than are life histories with a single type of density dependence. What is essential for a hydra effect is that mortality precedes or is concentrated in the early part of a strongly density-dependent stage.

Continuous models with structured life cycles

Temporal differences between mortality and density dependence in a continuous time model require some age- or stage-structure in the population. Two recent articles have examined models have assumed distinct density-dependent processes in each stage and specific functional forms for the density-dependent processes (De Roos et al. (2007) and Schreiber & Rudolf (2008)). Here, I use a generalization of Schreiber and Rudolf’s model to derive some common characteristics associated with hydra effects in continuous-time stage-structured models. Two ordinary differential equations describe the dynamics of juvenile and mature individuals. The first (juvenile) stage cannot reproduce. The population can experience density dependence in b, the per-capita birth rate of class 2 (adults, with population size N2). Alternatively or additionally, there may be density dependence in the per-capita maturation rate, g, of immature individuals (with population size N1). Both b and g are nonincreasing functions of both population densities. Birth rate, b, must at least decrease with adult density, while g must at least decrease with juvenile density. The model is:

image(7)

Overcompensation (reduced output with greater input) in adult reproduction is defined by the birth rate decreasing with greater adult population (N2b/∂N2 + b < 0), and overcompensation in maturation occurs when the number of individuals maturing per unit time decreases with larger numbers of juveniles (g + N1g/∂N1 < 0). The stages have density-independent per-capita death rates d1 and d2. The analysis determines how increasing either d1 or d2 (or both) alters the equilibrium populations of N1 and N2. Details are given in Appendix S3.

It is not necessary that both growth and birth exhibit density dependence for a hydra effect to occur. The reason why eqn 7 permits hydra effects while eqn  1 does not is that temporal separation of the action of mortality and density dependence is only possible in eqn  7. In fact, having density dependence in both stages generally reduces a hydra effect, because the increase in population size produced in one stage is diminished in the other. This is why the hydra effects in the examples presented in De Roos et al. (2007, Figs 1 and 2) only occur when mortality is a very small fraction of the maximum mortality (a fact obscured by the logarithmic scaling of their x-axes).

The basic results of the analysis are: (1) juvenile density always decreases with juvenile mortality; (2) juvenile density can increase with adult mortality when the reproductive rate function is overcompensatory with respect to adult density; (3) adult density always increases with juvenile mortality when the growth function is overcompensatory and (4) adult density can increase with adult mortality when the growth function is overcompensatory. The first case is the only one where a stage-specific hydra effect cannot occur. This is the one case where density-dependent growth operates simultaneously on the same variable (juvenile density) as the density-independent death rate. The other cases involve temporal separation of density-dependent input to a stage and density-independent loss from that stage. Independent overcompensatory density dependence in each stage produces offsetting changes in the output from each stage, making the conditions for an increase in the total population size with mortality more restrictive than when only a single stage has overcompensatory density dependence. However, overcompensation in both stages is frequently inconsistent with population dynamical stability, which was assumed in the analysis in Appendix S3. In addition, Schreiber & Rudolf’s (2008) analysis of a model like eqn  7 revealed that alternative attractors occurred frequently given low mortality rates; each attractor entailed overcompensatory density dependence in a different stage. Thus, the equilibrium analysis in Appendix S3 does not tell the full story.

In summary, mortality that operates prior to overcompensatory density dependence may produce hydra effects that are significant in magnitude, whether a population is modelled in discrete or continuous time. However, the exact parameter ranges producing hydra effects must be determined numerically for systems that lack a stable equilibrium point. The next section examines the third and final mechanism by which hydra effects can be produced.

Mechanism 3: consumer mortality leads to more prudent resource exploitation

Density dependence usually arises at least partially from resource depletion (Begon et al. 2006). Traditional models of density dependence are not sufficiently flexible to represent the diversity of feedback effects in consumer–resource interactions (Abrams 2002). This section will consider how hydra effects in consumer populations arise via interactions with resources when mortality reduces the consumer’s attack rates, and thereby increases resource productivity. Most work on this subject has dealt with continuous-time models. Because of the large amount of published work dealing with the mechanism, this section reviews previous work (Abrams & Vos 2003; Matsuda & Abrams 2004; Abrams & Matsuda 2005; Abrams & Quince 2005), rather than analysing particular models. It should be noted that all of the papers referred to here assume continuous-time systems.

The initial decrease in the population of a species brought about by increased mortality causes increases in other components of fitness due to compensatory changes in the density and/or characteristics of resources (prey), and other species in the food web (such as the resources’ own resources). Many resource-mediated changes have the potential to reverse the initial decline, producing an eventual increase in equilibrium or average population density (Abrams 1992, 2002, 2005; Abrams & Vos 2003; Matsuda & Abrams 2004; Abrams & Matsuda 2005). The underlying mechanism in all of these cases is reduced overexploitation of resources. It is well known that consumer population size often increases when its attack rate of resources decreases (Case 2001; Abrams 2002). Mortality can directly cause consumers to reduce their foraging (e.g. Matsuda & Abrams 2004). Mortality can also shift the balance of the predator’s foraging costs and benefits due to a greater resource (prey) population density, and this may also reduce the predator’s feeding rate. Hydra effects can be produced by type-2 consumer functional responses (Abrams 2002) because of the reduction in attack rate due to increased handling time with greater prey abundance. The association between instability and the hydra effect that exists in the simplest consumer–resource models is no longer present in systems with stage-structured resource (prey) populations (Abrams & Quince 2005), adaptive adjustment of costly defence by the prey (Abrams & Matsuda 2005) or two prey species differing in their vulnerability to the predator (Abrams & Matsuda 2005). Predator mortality can increase prey productivity, even with linear predator functional responses because of changes in the prey’s foraging (Abrams & Vos 2003), or changes in foraging or defence at still lower trophic levels (Abrams 1992). There is growing evidence for the dynamic importance and prevalence of adaptive antipredator behaviour by prey in ecological communities (Lima 1998; Bolker et al. 2003; Werner & Peacor 2003; Preisser et al. 2005).

Discussion

Review of theoretical results

An increase in population size as the result of increased density-independent mortality (a hydra effect) can occur in a wide range of circumstances. Such ‘hydra effects’ can arise in many of the commonly used models of density-dependent population growth and of consumer–resource interactions. They may occur in larger food webs as well. Hydra effects may apply to an entire population or some stage or age class within the population. Three different mechanisms for such effects have been distinguished here. The first mechanism is altered population fluctuations. In the most common scenario, increased mortality decreases fluctuations, resulting in a higher average density. However, the analysis here makes it clear that increased mortality does not always reduce fluctuations, and changes in the form as well as the amplitude of fluctuations are important. Many species appear to undergo cycles (Kendall et al. 1998), although the response of those cycles to mortality is seldom studied outside the laboratory (Dennis et al. 1997).

The second mechanism involves temporal differences between the action of density-independent mortality and the density-dependent processes that counteract it. This was responsible for the hydra effect in Ricker’s (1954) analysis of his discrete generation population, and it happens in stable as well as unstable systems. For this mechanism to increase the equilibrium density of a stable population, it is necessary that the density dependence be overcompensatory (decreased output of individuals from a stage as input increases). This is the reason why mortality before density dependence can produce a hydra effect in Ricker’s (1954) model but not that of Beverton & Holt (1957). Temporal separation of the action of density-dependent processes and density-independent mortality also occurs when density-dependent processes differ between, or only occur in one or a few age-classes or stages. Evidence for overcompensation is considered below.

The third basic mechanism producing hydra effects arises when mortality of consumers directly or indirectly leads to more ‘prudent’ (sensuSlobodkin 1974) resource exploitation. This means that the resource consumption rate of either the focal species or some species at a lower level in its food web is reduced as a result of the increased mortality. Adaptive foraging by a predator, its prey, or the prey’s foods can all produce this effect if their foraging effectiveness declines as a direct or indirect consequence of the increased mortality of the predator.

Although this analysis has focused on the counterintuitive effects of mortality, immigration can also have counterintuitive effects that involve many of the same mechanisms (e.g. Holt 2002). For example, immigration of prey can stabilize a cycling predator–prey system, thereby decreasing the mean prey density (Holt 1983, 2002).

The magnitude of hydra effects

The magnitude of hydra effects varies greatly depending on the specific model. Abrams (2002) showed that the density of a predator at the point where harvesting causes a predator–prey system to become stable could be many times greater than the density (equilibrium or mean) of the predator in the non-harvested system. The impact of mortality in discrete generation models increases as that mortality becomes more concentrated in the period before (or near the beginning of) density dependence. More strongly overcompensatory density dependence allows larger hydra effects in such models. For example, the parameter values for the Maynard Smith model used in Fig. 1b, would have produced a 5.9-fold increase in population size if the mortality was shifted to occur before, rather than after density dependence. [Correction added after online publication, 12 March 2009: 5.9-fold increase instead of 16.65; ‘mortality’ instead of ‘density dependence’] Very large magnitude hydra effects may occur when mortality causes a shift between alternative attractors, as in the two-stage model of Schreiber & Rudolf (2008). The strength of hydra effects based on the third mechanism depends on the magnitude of the change in the per-capita attack rate with a change in predator mortality, and also on the initial level of overexploitation of prey. Large changes in attack rates (e.g. with a large handling time; Abrams 2002), or strong initial overexploitation of prey, permit hydra effects that increase the density several-fold. When a type-2 functional response causes the decrease in resource capture rate, the hydra effect frequently characterizes most of the range of potential mortality rates (Abrams & Quince 2005).

In most of the systems with hydra effects that were considered here, increased mortality reduces the maximum per-capita growth rate at the same time that it increases population size. This suggests that hydra effects would be reduced by environmental fluctuation. There are many ways to incorporate environmental fluctuation, and a complete analysis cannot be given here. Numerical analysis of several models suggests that environmental fluctuations must be large to greatly reduce the magnitude of the effect, and those fluctuations typically have little effect on the range of mortalities over which a hydra effect occurs. Figure 4 presents some results demonstrating both of these conclusions for the Maynard Smith model (eqn  3).

Figure 4.

 The impact of environmental variation on the relationship between mean population size and fraction harvested. The parameters are identical to those in Fig. 3b. The solid line is the case with no environmental variation; the short-dashed line is a case in which B is chosen randomly at each time step from a uniform distribution between 0.5B and 1.5B. The long-dashed line is the same for a uniform distribution between 0.01B and 1.99B, which is very close to the maximum range allowable when the distribution is uniform.

When is density dependence likely to be overcompensatory?

Because overcompensatory density dependence plays a large role in hydra effects, it is useful to review some of the mechanisms by which it arises. Beverton & Holt’s (1957) derivation of their discrete generation model of population growth was based on the continuous and simultaneous action of density dependence and density-independent mortality. The lack of temporal separation of these processes is what prevented overcompensatory density dependence. Overexploitation of self-reproducing resources is a major mechanism by which overcompensation can arise. A theoretical study by Geritz & Kisdi (2004) showed that the overcompensation required for a hydra effect may arise from continuous consumer–resource interactions within generations in populations characterized by discrete generations. They showed that self-reproducing resources usually lead to the unimodal recursion relationships required for a hydra effect to operate when mortality precedes density dependence. Schreiber & Rudolf’s (2008) two-stage model assumed that the density dependence in each stage was due to depletion of a logistically growing resource. Overexploitation of at least one resource when mortality rates were low was required for overcompensation in that stage. Cannibalism and size thresholds for developmental transitions can also lead to overcompensation in stage transitions.

Relevant empirical evidence for overcompensation and hydra effects

Overcompensatory density dependence can lead to hydra effects, and it has been shown to be present in several systems. Nicholson (1950, 1954, 1957) showed that overcompensation occurred in laboratory populations blowflies, Lucilia cuprina. Additional evidence for this species was obtained by Moe et al. (2002) using a toxin (cadmium) to increase mortality. Postma et al. (1994) obtained similar results applying cadmium to a chironomid species. Overcompensation seems likely in northern populations of the Atlantic silverside, Menidia menidia (Munch et al. 2003); too large a juvenile population causes slow growth, resulting in very high mortality during the winter preceding maturation. Cannibalism was the main driver of overcompensatory density dependence in the Tribolium system studied by Dennis et al. (1997).

Given the abundance of evidence for conditions that could produce hydra effects, direct evidence for their existence is surprisingly rare. It is perhaps not so surprising for those cases driven by changes in population cycles, given the difficulty of maintaining cycling populations in the lab over long time periods, or maintaining field censuses over a long enough period to determine an accurate average density. However, some of the blowfly experiments of Nicholson (1950, 1954, 1957), which involved large amplitude population cycles, suggest that mean densities may have increased in response to mortality. On the other hand, Nicholson (1954) only reported decreases in adult density in response to various levels of mortality imposed on emerging adult flies in a population where egg production rate was density dependent. Cameron & Benton (2004) found that increasing egg mortality could increase adult density. This is consistent with the predictions of the two-stage model considered here, although it would not qualify as a stage-specific hydra effect. De Roos et al. (2007) interpret some early experiments on Daphnia by Slobodkin & Richman (1956) as evidence of an increase in abundance of the youngest age class due to harvesting, but Slobodkin & Richman (1956) themselves reached the opposite conclusion.

Two recent studies provide stronger support for the concept. A field study of smallmouth bass populations (Zipkin et al. 2008) found that the removal of all stages of the population in a single lake over 7 years resulted in an increase in population size, mainly due to an increase in the abundance of immature individuals. Pardini et al. (2009) quantified density dependence in the biennial invasive plant, garlic mustard. Their three-stage model of density-dependent growth with parameters estimated from field experiments exhibited two-point cycles that were qualitatively consistent with field observations. Pardini et al. (2009) found fairly broad ranges of mortality rates in which the population increased (relative to the unharvested population size) in response to the mortality. The plant literature also has some examples of a lack of response of population size to mortality (e.g. Newingham & Callaway 2006).

Not all cases of increase in response to mortality qualify as hydra effects. There is a large literature on increases of plant biomass in response to herbivory, following the studies by Paige & Whitham (1987) and Paige (1992). However, this often involves increased size of individual plants that have been grazed, and it must involve mechanisms that differ from the population-level effects discussed here.

Explaining this scarcity of examples of hydra effects remains a challenge. However, lack of awareness of the possibility of such effects is likely to be a contributing factor. Lack of awareness may have been promoted by the failure of previous modelling studies to cite previous work that would have argued for greater generality of the results. None of the papers reviewed here cited the original results of Ricker (1954). Studies using continuous time models have generally failed to cite those based on discrete time models, and vice versa. Different mechanisms have not been properly distinguished. Different terminology has been proposed in successive studies; De Roos et al. (2007) misconstrued the term ‘hydra effect’ to apply only to cyclic predator–prey systems, and proposed ‘overcompensation in biomass’ instead. This has the disadvantage that it implies a different meaning than the term (overcompensation) has in describing density dependence. Seno (2008) simply refers to the phenomenon as ‘the paradoxical effect’, which seems too broad. It is hoped that the present work will promote understanding of the generality of this phenomenon, whatever terminology is adopted.

Relevance to fisheries management

A large part of the literature dealing with the impacts of mortality has concerned fisheries management, and a major figure in fisheries biology was the first to note the possibility of a hydra effect (Ricker 1954). In subsequent fisheries research, there has been a persistent dispute about the existence of a ‘reproductive surplus’ that may be harvested without any impact on the population (discussed in Hilborn et al. 1995). Because the extreme position in this debate has been that such a surplus exists for most marine fish stocks, the possibility that harvesting may actually increase stock size has been largely overlooked. The implications of the hydra effect for fisheries are discussed in more detail elsewhere (Abrams & Matsuda 2005; Abrams & Quince 2005; De Roos et al. 2007). In one sense, the lack of awareness of Ricker’s (1954) idea in fisheries may have been beneficial, because the possibility of a hydra effect might have been used to justify greater harvesting. However, the models underlying hydra effect also predict that the harvest rate that maximizes the standing population is usually only slightly less than that which (if maintained) would result in a dramatic collapse or extinction. The harvest rate that maximizes yield is even closer to the collapse point. When population growth parameters are poorly known, this possibility argues for a highly precautionary approach.

Acknowledgements

This work was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. I thank Elise Zipkin, Robert Holt, James Grover and two anonymous referees for comments on previous drafts.

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