Michael B. Bonsall, Department of Zoology, University of Oxford, Oxford OX1 3PS, UK. Tel.: 01865 281064; fax: 01865 310477; e-mail: firstname.lastname@example.org
Parental care is of fundamental importance to understanding reproductive strategies and allocation decisions. Here, we explore how parental care strategies evolve in variable environments. Using a set of life-history trait trade-offs, we explore the relative costs and benefits of parental care in stochastic environments. Specifically, we consider the cases in which environmental variability results in varying adult death rates, egg death rates, reproductive rate and carrying capacity. Using a measure of fitness appropriate for stochastic environments, we find that parental care has the potential to evolve over a wide range of life-history characteristics when the environment is variable. A variable environment that affects adult or egg death rates can either increase or decrease the fitness of care relative to that in a constant environment, depending on the specific costs of care. Variability that affects carrying capacity or adult reproductive rate has negligible effects on the fitness associated with care. Increasing parental care across different life-history stages can increase fitness gains in variable environments. Costly investment in care is expected to affect the overall fitness benefits, the fitness optimum and rate of evolution of parental care. In general, we find that environmental variability, the life-history traits affected by such variability and the specific costs of care interact to determine whether care will be favoured in a variable environment and what levels of care will be selected.
Empirical work suggests that environmental variability affects parental care decisions within species. For example, environmental variability affects parental care decisions in kestrels (Gaibani et al., 2005) and cichlids (Townshend & Wootton, 1985). Likewise, Welham & Beauchamp (1997) found that environmental variability is expected to affect provisioning in black terns. Theoretical work by Carlisle (1982) suggests that environmental variation will affect intraspecific patterns of care. Although environmental variation is predicted to affect patterns of care once care has evolved in a species, the role of environmental variability (e.g. through resource heterogeneities) on the conditions that drive the origin of parental care from a state of no care remains unclear.
To begin to understand the role of environmental variation on the evolution of parental care, we develop a mathematical model of parental care and compare the evolution of parental care when the environment is constant vs. variable. As parental care is likely to be associated with various costs and benefits (Townshend & Wootton, 1985; Clutton-Brock, 1991), we specify a set of allocation and trade-off strategies to explore how various costs and benefits influence the evolution of parental care in constant and stochastic environments. We consider environmental variability in a very general manner. Specifically, we simply assume that individuals within a population experience a fixed number of environments that affect key life-history characteristics (i.e. mortality and reproductive rates). We then ask whether and under what conditions parental care will be favoured given the variability associated with environments experienced by that population. Our approach can apply to a population experiencing either spatial or temporal variability (discussed further below), and we do not specify how long individuals spend in each environment. The strategy (i.e. parental care) experiences all environments equally. Obviously, specific details of plasticity and predictability will further affect the evolution of parental care in complex ways, but this is not the focus of our current study. Rather, we identify the conditions under which parental care is favoured from an ancestral state of no care when the environment varies regardless of the precise mechanism that gives rise to such variability. Our results thus provide an exploration of the general conditions likely to favour the evolution of care in constant vs. variable environments.
The framework for exploring the evolution of parental care in a stochastic environment follows that outlined in Klug & Bonsall (2007, 2010). We establish an ecological interaction in which a rare mutant with a unique life-history strategy (i.e. parental care) is allowed to invade a resident population that exhibits no care.
We model a population in which groups of individuals experience some fixed number of environments that vary. Each environment, φ, is associated with a particular set of demographic and ecological characteristics that determine egg and adult mortality, the rate of egg fertilization and population carrying capacity (Fig. 1a, b). Such variation across environments could occur through either spatial or temporal heterogeneity in environmental quality (Fig. 1a, b). Particular ecological correlates or selective pressures associated with spatial vs. temporal variation will affect the dynamics and evolutionary outcomes (e.g. local adaptation, plasticity). This, however, is not something we explore in this study. Rather, our framework considers variability more generally: for a given level of variability in demographic or ecological processes, we identify the behavioural strategies that are likely to be selected for across all environments in a given population. In doing so, we identify the baseline conditions likely to favour care in fixed vs. stochastic environments.
In a given environment φ, individuals pass through egg (E), juvenile and adult (A) stages such that:
where r is the rate of egg fertilization (population growth rate), dEφ is the egg death rate and dAφ is the death rate of adults in a given environment, φ. mE is the egg maturation rate, τ is the time delay associated with juvenile development and σJ is the juvenile through-stage survival. Population growth is assumed to follow a linear density-dependent process, where K(φ) represents the population carrying capacity in the φth environment.
For each set of parental care trade-offs (Table 1), we assume that fertilization rate, egg or adult mortality, or population carrying capacity vary because of stochastic effects. In each set of analyses, we allow only one trait to vary across environments, and any difference in adult and egg densities across environments is then the result of variation in the trait of interest and the associated trade-offs linked to the evolution of parental care. For simplicity, other life-history traits (juvenile survival, juvenile development time and maturation rate) do not vary systematically across environments within a population. As we consider groups of individuals experiencing variability through demographic and ecological processes in a single trait (i.e. egg or adult survival or fertilization rate), it is not necessary to include explicit details that account for movement across environments or the amount of time individuals spend in each environment. This framework is consistent with a range of scenarios, including the case in which individuals remain in a given environment or the scenario in which all individuals experience each environment for some period of time.
Table 1. Nonlinear trade-offs associated with initial egg allocation and parental care. Egg survival (1-dE0) is a proxy for quality and hence initial parental egg allocation.
Baseline egg allocation strategies (residents)
Adult death rate (dA)
Adult death rate (dA) ↑ as egg survival (1-dE) ↑
Fertilization rate (r)
r = r0·exp (-(1-dE0))
Fertilization rate (r) ↑ as egg survival (1-dE) ↓
Parental care & egg allocation strategies (mutant strategy)
Egg death rate (dE) – Fertilization rate (r)
dEm = dE0 exp(-c), rm = r0·exp(-(1-dE0) + c)
Egg survival (1-dE) ↑ as care (c) ↑, Fertilization rate (r) ↓ as care (c) ↑ & egg survival (1-dE) ↑.
Egg death rate (dE) – Fertilization rate (r) – Adult death rate (dA)
dEm = dE0 exp(-c), rm = rA0·exp(-(1-dE0) + c),
Egg survival (1-dE) ↑ as care (c) ↑, Fertilization rate (r) ↓ as care (c) ↑ & egg survival (1-dE) ↑, Adult death rate (dA) ↑ as egg survival (1-dE) ↑ & care (c) ↑
Egg death rate (dE) – Adult death rate (dA)
dEm = dE0 exp(-c),
Egg survival (1-dE) ↑ as care (c) ↑, Adult death rate (dA) ↑ as egg survival (1-dE) ↑ & care (c) ↑
Fitness measure in a variable environment
The dynamics of a rare mutant that has a different life-history strategy invading a resident strategy (that is assumed to be in equilibrium in the φth environment) are described by the following equations and the incorporation of various trade-offs (discussed below and outlined in Table 1):
where the subscript m denotes a mutant strategy in environment φ and A* denotes the equilibrial abundance of the adult resident strategy also in environment φ. As the mutant is assumed to be rare, any density dependence operating occurs only through direct competition with the resident strategy in environment φ. This assumption allows us to focus on the early evolution of care by exploring the origin of care from an ancestral state of no care.
Elsewhere (Appendix of Klug & Bonsall, 2007), we have thoroughly explored the dynamics of our general modelling framework in a single environment and shown that over wide regions of parameter space that the dynamics are stable. Instability only occurs if the time delay associated with juvenile development is very large, particularly if the rate of egg fertilization is also high (Appendix of Klug & Bonsall, 2007).
The derivation of fitness in a stochastic environment begins by defining a matrix representing a rare mutant with a unique life-history strategy (i.e. parental care) in environment φ:
The expected fitness (λ) associated with parental care in this environment can be found by taking the determinant of the matrix and solving the resulting characteristic equation. The appropriate measure of fitness is then the dominant eigenvalue associated with the matrix. We extend this idea to multiple environments by assuming that carrying capacity, adult death rate, egg death rate or egg fertilization rate is a function of the different environments (φ) and define a block diagonal matrix (F):
where the entries along the diagonal of this matrix are the matrices (eqn 5) for the rare mutant strategy in each of the different environments (φ). Thus, we evaluate the fitness of a novel strategy by evaluating the strategy invasion dynamics in all the different environments within the population. The overall measure of fitness associated with the novel care strategy is then the dominant eigenvalue () of our φ by φ block diagonal matrix (eqn 6). In a variable environment, the calculation of fitness can be based on maximizing either long-term growth rate or the geometric mean fitness (Hastings & Caswell, 1979). Regarding the first, the dominant eigenvalue () is the expected long-term growth rate of the novel strategy that can be maximized according to some optimization criteria. Regarding the second approach, Lewontin & Cohen (1969) show that the expectation of the logarithm of the growth rate (E[ln()]) is the logarithm of the geometric mean (ln[G ()]). By expanding the logarithm of fitness about the mean (μ):
So the expectation of the logarithm of the growth rate is:
and the geometric mean fitness [G ()] is then:
By approximating e-x ≈ 1 −x:
where is the variance in the fitness measure.
Variance in fitness
The variance of a random variable can be determined from the difference between the expected value of the sum of squares and the expected value of the square of the sum () and in a stochastic environment, the variance in fitness equates to:
Trade-offs associated with parental care and initial egg allocation
Parents can improve offspring survival by investing energy and/or nutrients into eggs (referred to as initial egg allocation) and/or by providing post-fertilization parental care behaviour (referred to as parental care). Similar to Klug & Bonsall (2010), we use a set of nonlinear trade-offs to reflect costs and benefits associated with initial egg allocation and parental care (Table 1). We focus on nonlinear trade-offs as they are biologically realistic (and supported by empirical observations – e.g. Stearns, 1994) and have been used in previous models of parental investment and care (e.g. Smith & Fretwell, 1974; Sargent et al., 1987; Winkler, 1987; Klug & Bonsall, 2010), and because they allow for a broader exploration of parameter space and hence a more thorough exploration of the costs and benefits of care. Specifically, using nonlinear, asymptotic trade-off functions (Table 1) allows us to consider all biologically realistic parameter values (e.g. death and maturation rates between zero and one). In contrast, if we used linear trade-offs, we would only be able to consider a truncated range of parameter space (i.e. only those values which gave rise to biologically sensible death and maturation rates). Nonetheless, exploration of both linear and nonlinear trade-offs using this same general framework (Klug & Bonsall, 2007) suggests that linear vs. nonlinear trade-off functions are unlikely to affect our qualitative predictions.
Initial egg allocation is expected to affect both resident and mutant strategies. Egg survival (1-dE) is our proxy of initial egg allocation, and as such, adult death rate (dA) and/or fertilization rate (r) is nonlinear functions of egg survival (1-dE) (Table 1; see also Klug & Bonsall, 2010). Specifically, investing into eggs is costly to parents, such that as initial egg investment increases, adult death rate increases and/or reproductive rate decreases (Table 1).
The provision of parental care behaviour (c) introduces additional costs and benefits. Care acts to increase egg survival (i.e. offspring benefits; Table 1) but is costly to the parent in terms of opportunity for future reproduction as care acts to decrease adult survival and/or decrease fertilization rate (Table 1). As the full life-history consequences of parental care and initial egg investment are often difficult to discern (Clutton-Brock, 1991; Klug & Bonsall, 2010), we explore the strategies of parental care vs. no care under a range of different trade-off scenarios. In doing so, we identify the life-history trade-offs that are most likely to drive the evolution of parental care in a stochastic vs. fixed environment. We investigate the following trade-off combinations associated with the costs and benefits of parental care and initial egg investment: a strategy in which care and/or initial egg investment (i) increases egg survival but decreases parental fertilization rate (Table 1: dE-r trade-off), (ii) increases egg survival but decreases both adult fertilization rate and adult survival (Table 1: dE-r-dA trade-off) and (iii) increases egg survival but decreases adult survival (Table 1: dE-dA trade-off).
To analyse the effects of different life-history trade-offs, initial egg allocation strategies (Table 1) and environmental variability, we determine the optimal fitness () gains (under different trade-off strategies –Table 1) in variable and constant environments as the levels of parental care increase. As mentioned earlier, we introduce variability into several key traits (carrying capacity, K, egg death rate, de, adult death rate, dA, and fertilization rate, r) in the presence of different trade-offs (Table 1). To determine the strength of selection for a given combination of life-history trait trade-offs, we take the derivative of the geometric fitness function with respect to the levels of parental care. This measure of selection is based on the magnitude and the slope associated with selection on care: if the magnitude of this derivative is small, selection is weak and if the magnitude is large, selection is strong. Further, changes in the force of selection with respect to care give an indication of fitness optima (where the slope of with respect to care (c) is equal to zero).
Baseline dynamics: variability in carrying capacity
Irrespective of the underlying trade-off (Table 1), variability in carrying capacities across environments [K(φ)] has negligible effect on the evolution of parental care (Fig. 2). From eqn (5), the partial derivative of the function of egg population growth with respect to mutant adult density is where . As such, if the resident and mutant experience the same variability in carrying capacity, K(φ) cancels in this partial derivative. Although care increases fitness benefits for all trade-off scenarios (Fig. 2a–c), variability in carrying capacity does not lead to differential fitness effects (or associated costs) of care. However, details on the underlying trade-offs associated with care (Table 1) do affect the quantitative relationship between geometric mean fitness and levels of care (Fig. 2). Fitness benefits of care show either increasing (egg death rate – fertilization rate, egg death rate – adult death rate – fertilization rate trade-off) or diminishing (egg death rate – adult death rate trade-off) returns as care increases (Fig. 2). Further, only under the egg death rate, fertilization rate trade-off is a minimum investment in care (c >0) required before fitness benefits increase as care increases. This limiting case, where carrying capacity is unimportant in relation to environmental variability, also forms a baseline as the trade-offs affect the invasibility of the mutant strategy and the equilibrium density of the resident. This allows us to compare and contrast the detailed effects of different life-history trade-offs and variable demographic rates on the fitness effects associated with the evolution of care.
Costs and benefits of care: egg death rate – fertilization rate trade-off
When parental care increases egg survival but decreases future parental reproduction (i.e. adult reproductive rate; Table 1), increasing levels of care can lead to higher fitness gains (Fig. 3). In general, across different life-history traits under this trade-off scenario, increased variability reduces fitness (Fig. 3). More specifically, if variability acts through egg death rate, fitness gains increase with increasing parental care (Fig. 3a). Under highly variable egg death rates, fitness benefits only become substantial when high levels of care are provided (Fig. 3a). Environmental variability on fertilization rate has no discernable impact on differential fitness benefits leading to constant fitness benefits for a given level of care across different environmental variability regimes (Fig. 3b). Similar to egg death rate, variability in adult death rate across environments leads to different fitness benefits for different levels of care depending on the magnitude of the variability (Fig. 3c).
Patterns in the force of selection reveal that as investment in care increases, the strength of selection favouring care also increases (Fig. 4a). However, patterns of selection vary across environmental conditions. In constant environments, the force of selection is always an increasing function of care. In more variable environments, the force selection first decreases and then increases as care increases when egg death rate is variable. This leads to a minimum where the selection gradient on care falls to zero , and the point of this minimum for the investment in care (c) depends on the level of environmental variability on egg death rate (Fig. 4a). In contrast, variability on adult death rate has little effect on the force of selection, and the force of selection increases as the level of care increases when environmental variability affects adult death rate (Fig. 4a).
Costs and benefits of care: fertilization rate – adult death rate – egg death rate trade-off
When parental care increases egg survival but decreases parental fertilization rate and adult survival, fitness increases as the levels of care increases (Fig. 5). Minimum investment in care (c >0) is necessary before fitness benefits associated with care occur. Fitness associated with care decreases as variability in egg death rate increases (Fig. 5a) and, to a lesser extent, as variability in adult death rate increases (Fig. 5c). As in the previous trade-off scenario (Fig. 3b), environmental variability on fertilization rate has no discernable impact on differential fitness benefits leading to constant fitness benefits for given levels of care across different environmental regimes (Fig. 5b).
In general, the force of selection on investment in care is an increasing function of care. Variability in both egg death rate and adult death rate leads to minima in the force of selection and hence evolutionary optima where selection is weak. Under greater variability, this minimum occurs at a higher level of investment in care. However, these are not evolutionary stable points and are under disruptive selection because small increases or decreases in care lead to positive increases in the force of selection (Fig. 4b). In contrast, variability on fertilization again has little effect on the force of selection associated with care (Fig. 4b).
Costs and benefits of care: egg death rate – adult death rate trade-off
When care increases egg survival but decreases adult survival (Table 1), fitness benefits increase and then asymptote as care increases (Fig. 6) and this is in contrast to the previous trade-off scenarios. As in the previous trade-off scenarios, fitness gains only occur once a minimum threshold in care is achieved (c >> 0). Variability in baseline demographic rates leads to different effects of care on fitness. High variability in baseline egg death rate is expected to most strongly decrease the fitness benefits associated with care (Fig. 6a), whereas variability on fertilization rate has no discernable impact on fitness benefits (Fig. 6b), leading to constant fitness benefits for given levels of care. Interestingly and in contrast to changes in egg death rate, increasing variability in adult death rate is expected to lead to increased fitness as care increases (Fig. 6c).
Under variability in both egg death rate and adult death rate, the force of selection associated with this trade-off scenario declines as the investment in care increases (Fig. 4c). This reflects the diminishing fitness benefits observed under this type of trade-off when investment in care is high.
Environmental variability that affects life-history traits such as adult or egg survival impacts the conditions under which parental care will be favoured from an ancestral state of no care. However, the specific effect of environmental variability depends on the costs of care and the life-history traits affected by environmental variability. In many cases, environmental variability reduces the fitness gains, and hence selection for parental care particularly when the costs of parental care are associated with both reduced parental survival and reproductive rate (e.g. Figs 3a, c, 5a, c and 6a). These findings are consistent with the hypothesis that variable environments will often be associated with a decreased need for parental care (Stearns, 1976). In contrast, if the only cost of care is reduced parental death rate, environmental variability that affects adult death rate increases the fitness associated with care (Fig. 6c). This pattern is consistent with both the hypothesis that unpredictable environments can increase the fitness benefits of parental care in many animals (Clutton-Brock, 1991) and the prediction that low adult survival can favour the evolution of parental care (Klug & Bonsall, 2010). Additionally, environmental variability that affects adult reproductive rate does not affect selection on parental care in our model. Together, these findings illustrate that the conditions favouring the evolution of parental care are complex: the degree of environmental variability, the life-history traits that are affected by such variability and the specific costs of parental care interact to determine the strength of selection on parental care. Nonetheless, our findings suggest that parental care can be favoured over a broad range of life-history conditions in both variable and constant environments.
Depending on how the costs and benefits of egg investment and care are manifest, the fitness gains of various levels of parental care, and hence the force of selection, differ under alternative environmental scenarios. Thus, environmental variability is expected to affect the level of care that is selected for in the early evolution of parental care. For instance, combined effects of costly care on adult survival and benefits of increased egg survival have diminishing fitness gains associated with care in stochastic environments, which can affect the fitness optima and level of care that will be favoured. In contrast, the costly effects of care on fertilization rate and adult survival and beneficial effects of care on egg survival do not lead to diminishing fitness gains of care. Increases in the variability in adult death rate are expected to lead to fitness benefits as care increases. This is consistent with the idea that parents facing higher mortality risks in variable environments should increase care (Carlisle, 1982). Only when the costs of care act on adult survival through the egg death rate, adult death rate trade-off [and not through the effects of variability on baseline adult survival that is independent of effort (Table 1)] is fitness expected to increase in stochastic environments.
In general, a range of parental care strategies (associated with various life-history trade-offs) under a range of environmental regimes are expected to give rise to equivalent fitness. As such, it is expected that the force of selection will be relatively weak over a broad range of investment patterns and environmental variability regimes. With broad, equivalent fitness surfaces and weak force of selection, it is expected that a range of parental care strategies will be observed in stochastic environments. Further, these predictions suggest that parental care and investment strategies between different environments may be substantially distinct such that both high and low investment patterns may be observed for the same species under different environmental regimes. In stochastic environments, such phenotypic variability associated with parental care may evolve as a ‘bet-hedging’ strategy. Bounded variance around optimal phenotypes can evolve because of the effects of stabilizing selection (Sasaki & Ellner, 1995). Furthermore, Seger & Brockmann (1987) suggest that bet-hedging strategies may be more common in relatively short-lived species and strategies. Similarly, here we show that in stochastic environments, the processes of parental care that reduces adult survival lead to increased fitness optima if the force of selection around the optimum is weak. As such, it is expected that bet-hedging strategies may be more influential in the evolution of parental care in shorter-lived species.
Plasticity in parental care behaviour in relation to environmental variability has been empirically observed in some animals. In a study on European kestrels, Gaibani et al. (2005) show that populations of different birds living in different environments alter the levels of care depending on environmental variability. Populations of kestrels with more variability exhibited less care at the beginning of the season but not later, and on average, both populations produced the same number of offspring (although less parents bred in the variable environment). In highly variable environments, only those parents that have the ability to adjust provision to offspring are expected to reproduce and engage in parental care (Carlisle, 1982). Further, if environmental variability affects parental survival then parents should increase care, whereas if the mortality risks increase from parental effort (rather than the environmental variability) then the amount of care should decline (Carlisle, 1982).
In summary, we have highlighted a range of life-history trait trade-offs and variability regimes that are likely to influence the evolution of parental care in stochastic environments. Understanding how environmental variability affects life-history strategies such as the evolution of parental care requires a detailed understanding of how stochastic environmental regimes influence particular traits (Schaffer, 1974; Doak et al., 2005; Boyce et al., 2006). Together with related work on life-history evolution, the approach presented here will provide insights into the evolution of parental care and foster the development of more general approaches to the understanding of life-history evolution in variable environments.
This work was supported by the Royal Society (to MBB) and NSF International Research Program Fellowship # 0701286 (to HK). We are grateful to anonymous referees whose detailed comments improved this manuscript.