The strength of species interactions modifies population responses to environmental variation in competitive communities

Journal of Theoretical Biology 310 (2012) 199–205

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The strength of species interactions modifies population responses to environmental variation in competitive communities ¨ Lasse Ruokolainen a,n, Jorgen Ripa b a b

Department of Biosciences, University of Helsinki, Viikinkaari 1, P. O. Box 65, FI–00014 Helsingin yliopisto, Finland Theoretical Population Ecology and Evolution Group (ThePEG), Department of Biology, Lund University, S¨ olvegatan 37, SE-22362 Lund, Sweden

H I G H L I G H T S c c c c

Population stability in variable environments is studied in competitive communities. Population stability is sensitive to competition strength between species. Competition strength modifies species responses to environmental reddening. This is affected by density dependence and between-species environmental covariance.

a r t i c l e i n f o

abstract

Article history: Received 13 January 2012 Received in revised form 26 June 2012 Accepted 27 June 2012 Available online 7 July 2012

The life-history parameters of most living organisms are modified by fluctuations in environmental conditions. The impact of environmental autocorrelation on population persistence is well understood in single species systems. However, in multi-species communities the impact of stochasticity is complicated by the possibility of different species having differing intrinsic responses to the environment (environmental correlation). Previous work has shown that whether increasing between-species environmental correlation stabilises population fluctuations or not, depends on an interaction between density-dependence and environmental autocorrelation. Here we derive analytical conditions for how this interaction in turn depends on the strength of interspecific competition. Under relatively weak between-species interactions, increasing environmental autocorrelation always dampens population fluctuations, while increasing autocorrelation destabilises strongly interacting populations. In contrast, under intermediate interaction strengths, increasing autocorrelation destabilises (stabilises) population dynamics when populations respond independently (similarly) to environmental fluctuations. These results apply to a wide range of competitive communities and also have some relevance to consumerresource systems. The results presented here help us better understand population responses to environmental fluctuations under different conditions. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Autocorrelation Competition Density dependence Environmental correlation Stability

1. Introduction Temporal variation in environmental conditions is an integral part of biological systems. The manner in which variation influences the performance of individuals and populations clearly depends on the amplitude of the variation, but also on its temporal autocorrelation (also known as colour, in analogy with the spectrum of visible light), i.e., the predictability of change in environmental conditions. It is well known that unstructured, single-species populations respond differently to variation in environmental colour, depending on the nature of intrinsic population dynamics (Roughgarden, 1975a;

n

Corresponding author. E-mail address: lasse.ruokolainen@helsinki.fi (L. Ruokolainen).

0022-5193/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jtbi.2012.06.035

Ripa and Lundberg, 1996; Kaitala et al., 1997; Ripa and Heino, 1999). A general pattern is that increasing environmental autocorrelation (environmental reddening) destabilises populations with undercompensating growth and stabilises populations with overcompensatory dynamics (Roughgarden, 1975a; Ripa and Heino, 1999). An important implication of this is an increase (decrease) in population extinction risk. Short-term patterns in environmental colour tend to vary naturally over time [e.g., (Ruokolainen et al., 2009a)], and anthropogenic alteration in environmental characteristics are also possible [e.g., (Ruokolainen et al., 2009a; Wigley et al., 1998)]. Therefore, understanding how populations respond to variations in environmental conditions under different circumstances is important for predicting population viability in the future. In addition to intrinsic dynamics, the reaction of a population to environmental reddening can be further modified by the

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presence of interacting populations, such as competitors or predators (Ripa and Ives, 2003; Greenman and Benton, 2005a,b; Vasseur, 2007). In competitive communities, the correlation between species-specific environmental effects (environmental correlation; the way each species is affected by environmental fluctuations) has been identified as an important factor affecting community stability and persistence (Ripa and Ives, 2003; Roughgarden, 1975b; Doak et al., 1998; Ives et al., 1999; Ruokolainen et al., 2007; Ruokolainen et al., 2009b; Ruokolainen and Fowler, 2008). Assuming uniform environmental correlation across species pairs, a general result is that undercompensating population dynamics always results in decreased population stability with environmental reddening (Ruokolainen et al., 2009a). However, in the case of overcompensating population dynamics some studies have reported decreasing population stability under independent environmental effects and increasing stability under correlated environmental effects [e.g., (Ripa and Ives, 2003; Greenman and Benton, 2005b; Ruokolainen and Fowler, 2008)]. In contrast, other studies have found no qualitative difference between independent and correlated environmental effects (Ruokolainen et al., 2007; Ruokolainen et al., 2009b). A potential, yet rather unexplored factor affecting population stability in coloured environments is the strength of interspecific interactions. Recent results suggest that this might have a qualitative impact on community responses to environmental reddening (Ruokolainen et al., 2009b). Previous results indicate that under weak competition environmental reddening always stabilises populations with overcompensating growth (Ruokolainen et al., 2009a), whereas under relatively strong competition population responses to reddening are expected to be sensitive to the magnitude of between-species environmental correlation (Ripa and Ives, 2003). In this paper, we derive conditions for simple two-species competitive communities, under which the size of population fluctuations are expected to increase or decrease with environmental reddening. The analysis is based on analytical methods of finding population variances in stochastic conditions (Ripa and Ives, 2003; Greenman and Benton, 2005b; Roughgarden, 1975b; Johansson et al., 2010).

2. How does interaction strength modify species responses to environmental variation? 2.1. Population level variability We begin by considering the simplest possible community, a system with two competitors. The dynamics in this two-species competitive community follow the multi-species Ricker model [e.g., (Ripa and Ives, 2003; Kilpatrick and Ives, 2003)]:     X i,t þ aX j,t X i,t þ 1 ¼ X i,t exp r i 1 þ ei,t ð1Þ Ki where Xi,t is the density of species i at time t, r is the intrinsic population growth rate and K is the carrying capacity (both assumed identical for all species, and K ¼1), and a is the strength of interspecific competition. The variable e represents the environmental variation as a first order autoregressive process, affecting the per capita population growth. For multiple, potentially inter-correlated time series, this can be modelled as (Ruokolainen and Fowler, 2008)

ei,t ¼ kei,t1 þ s

b¼

sffiffiffiffiffiffiffiffiffiffi 1r

r

ðjt1 þ boi,t1 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1þb

ð2aÞ

ð2bÞ

where r is the correlation coefficient between all pairs of environmental time series (0r r r1) and k is the autocorrelation coefficient (9k9 o1). The terms jt and oi,t are standard normal random components, where the former is common for all species and the latter independent between species. The parameter b is a scaling factor ensuring that noise variance remains independent of r, and the parameter s stands for the desired standard deviation of the time series. The following analysis is based on linearising the above model around its deterministic equilibrium. In stochastic, coloured environments, the population variance– covariance matrix (with the variance of population densities on the diagonal and the covariance between-species population fluctuations on the off-diagonal) can be approximated by the following equation [Appendix D in Greenman and Benton (2005b)], provided that the system is not too close to the instability boundary (r ¼2): VecðCÞ ¼ ðIk2 B  BÞðIkI  BÞ1 ðIkB  IÞ1 ðIB  BÞ1 VecðSÞ

ð3Þ

where Vec(C) is the vectorised variance–covariance matrix; I is an identity matrix; B is the Jacobian matrix, and Vec(S) is a vectorised variance–covariance matrix for the environmental noise (diagonal elements equal to noise variances s2 and offdiagonal elements equal to rs2). The  symbol indicates the Kronecker tensor product (producing all possible combinations between the elements in two matrices). Eq. (3) is general but leaves little room for biological interpretations. An alternative, equivalent, method of finding the stationary variance and covariance of interacting species in correlated and autocorrelated environments was used by Ripa and Ives (2003) and later by Johansson et al. (2010). The method is based on a coordinate transform of population densities to the directions of the eigenvectors of the Jacobian matrix, calculating the variance and covariance of the transformed coordinates, and transforming back to natural coordinates (population densities) again. The advantage of the coordinate transform is that the dynamics of the eigenvector coordinates are independent and simple, dictated by the single corresponding eigenvalues. As an example, consider the Jacobian matrix of our system (Eq. (1)) " # 1 1 þr a  1rþaa B¼ ð4Þ  1rþaa 1 1 þr a B has eigenvalues l1 ¼1  r, and l2 ¼1 r(1  a)/(1þ a) with the corresponding eigenvectors v1 ¼[11]T and v2 ¼[1–1]T, respectively. The first eigenvector (v1) corresponds to synchronous fluctuations of the two populations (in the direction X1 ¼ X2). The corresponding eigenvalue (l1 ¼1  r) is positive for low rvalues, corresponding to slow, positively autocorrelated fluctuations in this direction, i.e. slow, synchronized dynamics of the two competitors. A high r, on the other hand, gives a negative eigenvalue and corresponding rapid, ‘boom-and-bust’, dynamics of the two populations—they are both highly abundant one year and rare the next, all due to strong intraspecific competition. The second eigenvector (v2) represents perfectly asynchronous fluctuations (deviations from equilibrium of equal magnitude but opposite sign). The corresponding eigenvalue (l2 ¼1  r(1 a)/ (1þ a)) is close to the first eigenvalue in the case of weak competition (a low a), but approaches 1 as competition becomes strong (a approaches 1). In other words, in a strongly competitive system there will be slow fluctuations of high amplitude in this direction. In the limit a-1, the equilibrium loses its stability and the dynamics in this direction becomes a random walk (the two populations become ecologically equivalent). (See Ripa and Ives, (2003) for a more detailed discussion on this.)

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The variance of a single population depends on the variability along both eigenvectors. First considering the case without environmental autocorrelation (k ¼0), i.e., assuming white noise, the single population variance is given by " # " # s2 1 þ r s2 1r V i 9k ¼ 0 ¼ þ ð5Þ 2 1l2 2 1l2 1

2

Eq. (5) has two terms, one per eigenvector/eigenvalue pair. In the general case, there would be an interaction term as well, but due to the symmetry of our model it equals zero, which simplifies interpretations. If present, the interaction term is small unless the two eigenvalues are similar and not large in magnitude (Ripa and Ives, 2003). From Eq. (5) we see that environmental correlation has opposite effects on the two terms, hence the total effect depends on the relative magnitude of the two eigenvalues, which in turn depends on the strength of competition, a (Fig. 1(a)). To figure out which value of a is associated with population variance 2 2 being independent of r, we need to solve l1 ¼ l2 with respect to a, which gives

a ¼ r21

ð6Þ

Varying a can thus have a qualitative effect on the influence of environmental correlation (r) only when population growth is overcompensatory (r 41). Under this assumption, if a or–1, the first eigenvalue is highest in magnitude and an increasing environmental correlation amplifies population variability. In contrast, if a 4r–1 the second term in Eq. (5) dominates and variability is reduced by increasing environmental correlation (Fig. 1(b)). The problem becomes somewhat more complicated if environmental variation is coloured, which is to be expected under natural conditions (Ruokolainen et al., 2009a; Vasseur and Yodzis,

Jacobian eigenvalue

1 0.5

λ2

0

λ1

-0.5 -1

log Population variance

0 -0.2

ρ =0

-0.4 -0.6 -0.8 -1

ρ = 0.9

-1.2 -1.4 0

0.25

0.5

0.75

1

Interaction strength, α Fig. 1. (a) The relative magnitude of Jacobian eigenvalues (l1, dashed lines; l2, solid lines) depends on the strength of between-species competition (a). Grey and black lines represent cases with r¼ 0.5 and r ¼1.5, respectively. (b) Increasing a affects population variance depending on whether species respond to environmental fluctuations independently (r ¼0) or in a correlated fashion (r ¼ 0.9). Noise is assumed to be white in colour (k ¼ 0). Population growth is assumed to be overcompensatory (r¼ 1.5), with K¼1. Environmental variance s2 ¼0.5.

201

2004). In this more general case, we get " #" # " #" # s2 1þ r 1 þ l1 k s2 1r 1 þ l2 k Vi ¼ þ 2 2 2 1l2 2 1l2 1l 1l 1

1

2

ð7Þ

2

which again simplifies to two terms, one per eigenvalue/eigenvector. The relative contribution of each term is here modified by an interaction with the environmental autocorrelation k. The interaction factor is amplifying if k and the corresponding eigenvalue are of equal signs. We assume that the environmental autocorrelation is always positive (k Z0), which implies that the factor (1þ lik)/(1  lik) increases monotonically with li (i¼1,2). Note that despite the nonlinear interaction between the eigenvalues and k, Eq. (7) is still linear with respect to the environmental correlation (r). An increase in r is stabilising if and only if " # #  "  1 1 þ l1 k 1 1 þ l2 k o ð8Þ 2 2 1l1 1l1 k 1l2 1l2 k If both eigenvalues are positive, which is the case when ro1, environmental reddening will always increase population variance, irrespective of environmental correlation (Fig. 2(a)). In other words, if intraspecific competition is weak, the population dynamics are always underdamped, and a positive environmental autocorrelation always amplifies the variability in population size. Moreover, the second eigenvalue will always dominate in magnitude (Fig. 1(a)). It follows that Eq. (8) is always fulfilled and an increased environmental correlation is always stabilising. If, on the other hand, intraspecific competition is strong (r 41), the sign of the first eigenvalue is negative but the sign of the second depends on the value of a (Fig. 1(a)). If interspecific competition is relatively weak, a o(r  1)/(rþ 1), both eigenvalues are negative and environmental reddening reduces population variability independently of environmental correlation (Fig. 2(b)). In addition, under white environmental variation population variance increases with increasing environmental correlation, whereas under reddened environmental fluctuations increasing r tends to be stabilising (the first eigenvalue dominates, which means Eq. (8) is not fulfilled unless k is sufficiently large) (Fig. 2(b)). A higher value of a implies that the two eigenvalues are of opposite sign and the effect of environmental reddening (an increased k) is negative on the first term and positive on the second. The total effect thus depends on the relative strengths of the two terms in Eq. (7), which depends on the magnitudes of the two eigenvalues as well as the environmental correlation (r) and autocorrelation (k). If a or  1 the first eigenvalue dominates and the first term is the largest as long as k is not too large. At higher values of k the second term might grow faster than the decline of the first, especially if r is low. In other words, if environmental correlation is low and interspecific competition is of intermediate strength, there will be an intermediate level of environmental autocorrelation, which minimises population variability (Fig. 2(c)). The effect of environmental correlation is the same as above, destabilising at low values of k, but stabilising if the environmental autocorrelation is strong enough (Eq. (8), Fig. 2(c)). At the boundary where the eigenvalues have equal magnitude (a ¼r  1), environmental reddening is always destabilising when species respond to the environment independently (r ¼0), and always stabilising when they do so in a strongly correlated fashion (r ¼0.9). Finally, at high values of a the second term in Eq. (7) will dominate, unless r is high. This means that in situations with strong interspecific competition, environmental reddening will increase population variability unless there is also a strong environmental correlation, in which case the response to environmental reddening is again non-monotonic (Fig. 2(d)). The effect

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r = 0.5, α = 0.2

1

r = 1.5, α = 0.15

1 -1.2

0.8 0.5

-1

Environmental autocorrelation, κ

0.6 0.4

-0.8 0

0.2

-0.6

0 r = 1.5, α = 0.4

1

r = 1.5, α = 0.8 1.5

-1.2

0.8 0.6

-1

1

-1

0.5

-0.8

-0.5 0

0.4

-0.6

0.2 0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Environmental correlation, ρ Fig. 2. There is an interaction between environmental reddening (k) and between-species environmental correlation (r) affecting population variance (Vi). This interaction further depends on the strength of population density dependence (r) and between-species competition (a). The grey-shaded area in panels (c) and (d) indicates that population variance increases with increasing k; i.e., Vi is minimised at the boundary between the white and grey areas. Panels show log(Vi) under environmental variance s2 ¼0.5.

of the environmental correlation itself is always stabilising (Eq. (8) is always fulfilled).

2.2. Effect of community size at population level The above analysis deals with the simplest possible community of two competitors. This simple case is readily interpreted based on the eigenvalues of the community Jacobian (Eq. (7)). Unfortunately this quickly becomes quite complex when community size is increased. However, if we assume diffuse competition between species (i.e., aij ¼ a), it can be reasoned that increasing community size (n) will have the same effect as increasing a in a two-species system. As a result, increasing n changes the relationship between environmental correlation (r) and autocorrelation (k) as described in Fig. 2(a)–(c), depending on the underlying, common interaction strength. While population variances in an n-species community can be found using Eq. (3), the qualitative interaction between n and a can be approximated by taking the derivative of population variances with respect to k. This remains only an approximation, as population variance can be a non-monotonic function of k under some conditions. The result nonetheless verifies that adding more species to the community effectively decreases the critical value of a, where environmental reddening becomes destabilising due to the cumulative increment of interaction pressure on each population (Fig. 3).

3. Discussion In simple competitive communities environmental reddening (increasing temporal autocorrelation) has a general influence on community-level dynamics (Ruokolainen et al., 2009a); biomass variability increases with reddening, if individual populations are undercompensatory in their intrinsic dynamics, whereas overcompensating population growth is associated with reduced biomass variability. The similarity of species-specific reactions to environmental fluctuations (environmental correlation) only modifies the strength of this relationship, such that increasing correlation between species-specific environmental effects increases biomass variability (Ives and Hughes, 2002). Here we show that while the strength of interspecific interactions is not important at the ensemble level, it can considerably affect the dynamics of individual populations. Assuming diffuse interspecific competition (all pairwise interactions are equal), there is an interaction between the intrinsic population growth rate (r), the strength of interspecific competition (a), and community size (n), in affecting population-level responses to environmental reddening. Most importantly, interaction strength has a qualitative impact on the relationship between environmental correlation (r) and reddening, under overcompensating population dynamics. While the results are based on very simple diffuse competitive systems, the qualitative predictions hold for a wide range of more complex competitive communities, as well as non-uniform, between-species environmental correlation structures.

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10

Community size, n

9 8 7 6

ρ = 0.9

ρ =0

5 4 3 2 0

0.25

0.5

0.75

1

Interaction strength, α Fig. 3. Increasing community size (n) reduces the critical level of interspecific interaction (a) associated with a qualitative change in the response of population variability (log of Vi) to environmental reddening. This effect does not have a qualitative dependence on the correlation between species environmental effects (r). The lines represent zero-contours of the derivative of log(Vi) with respect to increasing environmental autocorrelation (k). Parameters: s2 ¼ 0.1, r ¼1.5.

It is also possible that varying interaction strength qualitatively affects population responses to environmental reddening in association with other forms of between-species interactions (such as predation). Understanding how populations respond to variation in environmental colour–which can be due to both natural and anthropogenic factors [e.g., (Ruokolainen et al., 2009a)]–under different conditions is important for predicting population viability and stability. When there is no autocorrelation in the environment (white noise), whether population variability in competitive communities decreases or increases with increasing environmental correlation (r) depends on the interaction strength, a. When a is greater than r  1, population fluctuations are dampened by increasing environmental correlation (Fig. 1(b)). In coloured environments the sign and relative magnitude of the Jacobian eigenvalues indicates how environmental variation filters through population dynamics. Under reddened environmental variation population variability tends to be dampened by increasing environmental correlation (Fig. 2). The strength of interspecific interactions–affecting the magnitude of the second Jacobian eigenvalue l2–has an important role in modifying the qualitative impact of environmental reddening under different environmental correlation structures. If competition is relatively weak, environmental reddening dampens population fluctuations independently of environmental correlation (Fig. 2(a) and (b)). Under strong intraspecific competition and intermediate interspecific competition, there may be an intermediate level of environmental autocorrelation which minimises population variability. This level of environmental autocorrelation increases with the environmental correlation and decreases with the strength of interspecific competition (Fig. 2(c) and (d)). These findings clarify discrepancies in previous reports using different strengths for intra- and inter-specific interactions. For example, some studies have reported that environmental reddening is destabilising when species respond to environmental fluctuations independently (r ¼0) and stabilising under strong environmental correlation (r ¼ 0.9) (Ripa and Ives, 2003; Greenman and Benton, 2005b; Ruokolainen and Fowler, 2008), while other studies have found no qualitative effect of varying environmental correlation (Ruokolainen et al., 2007; Ruokolainen

203

et al., 2009b). This discrepancy stems from different levels of intra- and interspecific interaction strengths applied in the different studies. We have assumed that all between-species interaction terms are equal. While the distribution of pair-wise interaction strengths can have an effect on the dynamical stability of communities (Kokkoris et al., 2002; Jansen and Kokkoris, 2003; Fowler, 2010), a preliminary investigation of randomly generated, stable communities with either uniformly or a beta distributed interaction strengths (Fowler, 2010) did not reveal any qualitative effect of the distribution of species interaction strengths. Both distributions led to decreased (increased) mean population variability with environmental reddening under weak (strong) interactions. The analysis presented above assumed that both betweenspecies interactions and environmental correlations are equal across species pairs. A more realistic scenario is that betweenspecies interaction strength and environmental correlation depend on the degree of between-species niche separation across a gradient (Ruokolainen et al., 2009b; Lehman and Tilman, 2000; Hughes et al., 2002). A previous study of stochastic communities, with such ‘hierarchical’ structure in both species interactions and environmental correlations, indicated that the population-level response to environmental reddening is qualitatively sensitive to the strength of interspecific interactions (Ruokolainen et al., 2009b). Indeed, an analytical investigation of hierarchically structured communities verifies that the qualitative impact of interaction strength observed in diffuse communities is also present in ‘hierarchical’ communities (Fig. S1, in Supporting online material); population variability decreases (increases) with environmental reddening under relatively weak (strong) interactions, assuming overcompensating population dynamics. This result suggests that while the assumptions in the above analysis were overly simplified when considering the structuring of natural communities, the qualitative predictions derived under these simplified assumptions are likely to be rather general. Many models have predicted that the average interaction strength should be relatively low in order to complex communities to persist [e.g., (Fowler, 2010; McCann et al., 1998; Berlow, 1999)], which matches observations from natural systems (Wootton and Emmerson, 2005). Under this assumption, our results predict a qualitative change in population level dynamics under environmental fluctuations when community size is increased. In the smallest community (n ¼2) relatively low interaction strengths were associated with decreased population variability with environmental reddening for both independent (r ¼0) and correlated (r ¼ 0.9) species environmental responses (Fig. 3). However, increasing the community size eventually led to a change in dynamics under independent responses, such that population variability started to increase with reddening. As reddening can have a strong effect on between population synchrony (Greenman and Benton, 2005b; Ruokolainen and Fowler, 2008), changes in the population-level response to reddening could also have an impact at the community level when community size is increased. This is because more asynchronous fluctuations between populations tend to be evened out at the ensemble level (Doak et al., 1998; Tilman et al., 1998). Interestingly, under independent environmental responses relatively low environmental autocorrelation (k) is associated with increased population synchrony with increasing species number, while relatively high k results in an opposite relationship (Fig. S2a). When considering correlated responses, increasing community size is always associated with a decrease in population synchrony (Fig. S2b). The dynamics of consumer-resource systems in coloured environments has been studied far less than competitive communities. Ripa et al. (1998) demonstrated that consumer

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populations are generally more reddened in their dynamics than their resources are. In coloured environments both reddening and increasing environmental correlation tend to amplify a consumerresource cycle (Ripa and Ives, 2003). More recently Vasseur (2007) showed that consumers and resources can differ qualitatively in their response to environmental reddening, such that resource variability decreases while consumer variability increases with reddening. Due to the cyclic nature of the consumer–resource interaction, population variances tend to be maximised at some intermediate environmental autocorrelation resonating with the deterministic cycle (Ripa and Ives, 2003). The strength of the consumer–resource interaction modifies the period length of the consumer–resource cycle, which in turn determines the location of the variance maximum along varying k (Ripa and Ives, 2003). Preliminary investigation indicates that the strength of the consumer–resource interaction can shift the consumer and resource variance maxima with respect to each other, indicating that varying interaction strengths can have a broad ecological relevance when considering the impact of environmental reddening in communities. For evaluating the applicability of the results presented here it is important to ask which factors affect the strength of species interactions in natural communities. Unfortunately, most research on interaction strengths has focused on consumer-resource links (Berlow et al., 2004). From this body of work, potential factors controlling interaction strengths can be found. These factors can be divided into two broad categories; those stemming from internal food web composition and those due to external (ambient) conditions. There is a tendency for the strength of consumer–resource interactions to become stronger with increasing consumer:resource biomass ratios (Wootton and Emmerson, 2005) due to an increase in relative energy flux from resources to consumers (Rip and McCann, 2011). As the body size ratios of herbivores compared to primary producers are larger in aquatic ecosystems than in terrestrial ones, general differences in interaction strengths are expected between ecosystem types (Rip and McCann, 2011). Results from soil food webs suggest that consumer feeding rates decline higher up the food chain (de Ruiter et al., 1995), which can bring about variation in energy flux between consumer-resource links at lower and higher trophic positions. The effect of external conditions can be seen, e.g., in the reduction of plant palatability with increasing environmental stress, which consequently leads to increased herbivory, as reduced food quality needs to be compensated for (Pennings and Silliman, 2005). More generally, if consumption rates depend on ambient conditions, such as temperature, environmental changes can lead to alterations in interaction strengths [e.g., (Sanford, 1999; Englund et al., 2011)]. This study demonstrates that the way in which populations in competitive communities respond to environmental reddening does not only depend on intrinsic dynamics and environmental correlation, but also on the strength of interspecific competition [see also (Ruokolainen et al., 2009b)]. Furthermore, all three components interact in affecting the qualitative population response to reddening. This can be important when considering, e.g., the relationship between community diversity and stability (McCann, 2000), or when loss of species from a community leads to a significant change in the distribution of pair-wise interaction strengths within the remaining community. While preliminary investigations of a consumer-resource system also show a qualitative impact of varying interaction strength, more work on this topic is needed to better understand the potential impacts of environmental reddening in food webs. Multiple factors, both internal and external, create potential for variation in interaction strength both between and within food webs. The result outlined in this paper indicate that such variation can lead to differences in population responses to environmental fluctuations, consequently altering population extinction risk.

Acknowledgements We would like to thank, Mike Fowler, Veijo Kaitala, Per Lundberg, Bo Ebenman, Jouni Laakso, and an anonymous reviewer for valuable comments and discussions that helped improving this paper. This work was funded by the Academy of Finland (LR) and the Swedish Research Council (JR).

Appendix White environments In the main text it is shown that environmental correlation (r) has no effect on population variances when a ¼r  1. At this exact point the Jacobian eigenvalues (l1 and l2) have equal magnitude, such that:     2r 91r9 ¼ r þ1 1þa a ¼ r1 ðA:1Þ Under this condition dynamics in the community have equal magnitude along the two eigenvectors of the Jacobian—v1 ¼[1,1] and v2 ¼[1,  1], corresponding to variation in total community biomass (i.e., X1 þX2) and compensatory dynamics between species (i.e., X1–X2), respectively. Coloured environments To evaluate the effect of varying environmental correlation (r) in coloured environments (k a in Eq. (7) in the main text by solving for the difference between population variance under correlated environmental effects (r a in and independent environmental effects (r ¼0): 2

V i 9r a 0 V i 9r ¼ 0 ¼ s2 r

2

2

2

ðl1 l2 Þ½k2 ðl1 l2 þ l1 l2 Þ þ kðl1 þ l2 2Þl1 l2  2 2 2ðl1 1Þðl2 1Þð

kl1 1Þðkl2 1Þ ðA:2Þ

Eq. (A. 2) indicates that varying r has no effect on population variance either when l1–l2 ¼0, or when k2 ðl21 l2 þ l22 l1 Þ þ kðl21 þ l22 2Þl1 l2 ¼ 0. Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jtbi.2012.06. 035.

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