Stochastic persistence of interacting structured populations

# Persistence in fluctuating environments for interacting structured populations

Gregory Roth  and  Sebastian J. Schreiber Department of Evolution and Ecology, One Shields Avenue, University of California, Davis, CA 95616 USA
###### Abstract.

Individuals within any species exhibit differences in size, developmental state, or spatial location. These differences coupled with environmental fluctuations in demographic rates can have subtle effects on population persistence and species coexistence. To understand these effects, we provide a general theory for coexistence of structured, interacting species living in a stochastic environment. The theory is applicable to nonlinear, multi species matrix models with stochastically varying parameters. The theory relies on long-term growth rates of species corresponding to the dominant Lyapunov exponents of random matrix products. Our coexistence criterion requires that a convex combination of these long-term growth rates is positive with probability one whenever one or more species are at low density. When this condition holds, the community is stochastically persistent: the fraction of time that a species density goes below approaches zero as approaches zero. Applications to predator-prey interactions in an autocorrelated environment, a stochastic LPA model, and spatial lottery models are provided. These applications demonstrate that positive autocorrelations in temporal fluctuations can disrupt predator-prey coexistence, fluctuations in log-fecundity can facilitate persistence in structured populations, and long-lived, relatively sedentary competing populations are likely to coexist in spatially and temporally heterogenous environments.

## 1. Introduction

All populations are structured and experience environmental fluctuations. Population structure may arise to individual differences in age, size, and spatial location (Metz and Diekmann, 1986; Caswell, 2001; Holyoak et al., 2005). Temporal fluctuations in environmental factors such light, precipitation, and temperature occur in all natural marine, freshwater and terrestrial systems. Since these environmental factors can influence survival, growth, and reproduction, environmental fluctuations result in demographic fluctuations that may influence species persistence and the composition of ecological communities (Tuljapurkar, 1990; Chesson, 2000b; Kuang and Chesson, 2009). Here we present, for the first time, a general approach to studying coexistence of structured populations in fluctuating environments.

For species interacting in an ecosystem, a fundamental question is what are the minimal conditions to ensure the long-term persistence of all species. Historically, theoretical ecologists characterize persistence by the existence of an asymptotic equilibrium in which the proportion of each population is strictly positive (May, 1975; Roughgarden, 1979). More recently, coexistence was equated with the existence of an attractor bounded away from extinction (Hastings, 1988), a definition that ensures populations will persist despite small, random perturbations of the populations (Schreiber, 2006, 2007). However, “environmental perturbations are often vigourous shake-ups, rather than gentle stirrings” (Jansen and Sigmund, 1998). To account for large, but rare, perturbations, the concept of permanence, or uniform persistence, was introduced in late 1970s (Freedman and Waltman, 1977; Schuster et al., 1979). Uniform persistence requires that asymptotically species densities remain uniformly bounded away from extinction. In addition, permanence requires that the system is dissipative i.e. asymptotically species densities remain uniformly bounded from above. Various mathematical approaches exist for verifying permanence (Hutson and Schmitt, 1992; Smith and Thieme, 2011) including topological characterizations with respect to chain recurrence (Butler and Waltman, 1986; Hofbauer and So, 1989), average Lyapunov functions (Hofbauer, 1981; Hutson, 1984; Garay and Hofbauer, 2003), and measure theoretic approaches (Schreiber, 2000; Hofbauer and Schreiber, 2010). The latter two approaches involve the long-term, per-capita growth rates of species when rare. For discrete-time, unstructured models of the form where is the vector of population densities at time , the long-term growth rate of species with initial community state equals

 ri(x)=limsupt→∞1tt−1∑s=0logfi(xs).

Garay and Hofbauer (2003) showed, under appropriate assumptions, that the system is permanent provided there exist positive weights associated with each species such that for any initial condition with one or more missing species (i.e. ). Intuitively, the community persists if on average the community increases when rare.

The permanence criterion for unstructured populations also extends to structured populations. However, in this case, the long-term growth rate is more complicated. Consider, for example, when both time and the structuring variables are discrete; the population dynamics are given by where is a vector corresponding to the densities of the stages of species , , and are non-negative matrices. Then the long term growth rate of species corresponds to the dominant Lyapunov exponent associated with the matrices along the population trajectory:

 ri(x)=limsupt→∞1tlog∥Ai(xt−1)…Ai(x0)∥.

At the extinction state , the long-term growth rate simply corresponds to the of the largest eigenvalue of . For structured single-species models, Cushing (1998); Kon et al. (2004) proved that implies permanence. For structured, continuous-time, multiple species models, can be defined in an analogous manner to the discrete-time case using the fundamental matrix of the variational equation. Hofbauer and Schreiber (2010) showed, under appropriate assumptions, that for all in the extinction set is sufficient for permanence. For discrete-time structured models, however, there exists no general proof of this fact (see, however, Salceanu and Smith (2009a, b, 2010)). When both time and the structuring variables are continuous, the models become infinite dimensional and may be formulated as partial differential equations or functional differential equations. Much work has been done is this direction (Hutson and Moran, 1987; Zhao and Hutson, 1994; Thieme, 2009, 2011; Magal et al., 2010; Xu and Zhao, 2003; Jin and Zhao, 2009). In particular, for reaction-diffusion equations, the long-term growth rates correspond to growth rates of semi-groups of linear operators and, for all in the extinction set also ensures permanence for these models (Hutson and Moran, 1987; Zhao and Hutson, 1994; Cantrell and Cosner, 2003).

Environmental stochasticity can be a potent force for disrupting population persistence yet maintaining biodiversity. Classical stochastic demography theory for stochastic matrix models shows that temporally uncorrelated fluctuations in the projection matrices reduce the long-term growth rates of populations when rare (Tuljapurkar, 1990; Boyce et al., 2006). Hence, increases in the magnitude of these uncorrelated fluctuations can shift populations from persisting to asymptotic extinction. Under suitable conditions, the long-term growth rate for these models is given by the limit with probability one. When , the population grows exponentially with probability one for these density-independent models. When , the population declines exponentially with probability one. Hardin et al. (1988) and Benaïm and Schreiber (2009) proved that these conclusions extend to models with compensating density-dependence. However, instead of growing without bound when , the populations converge to a positive stationary distribution with probability one. These results, however, do not apply to models with over-compensating density-dependence or, more generally, non-monotonic responses of demography to density.

Environmental stochasticity can promote diversity through the storage effect (Chesson and Warner, 1981; Chesson, 1982) in which asynchronous fluctuations of favorable conditions can allow long-lived species competing for space to coexist. The theory for coexistence in stochastic environments has focused on stochastic difference equations of the form where is a sequence of independent, identically distributed random variables (for a review see  (Schreiber, 2012)). Schreiber et al. (2011) prove that coexistence, in a suitable sense, occurs provided that with probability for all in the extinction set. Similar to the deterministic case, the long-term growth rate of species equals . Here, stochastic coexistence implies that each species spends an arbitrarily small fraction of time near arbitrarily small densities.

Here, we develop persistence theory for models simultaneously accounting for species interactions, population structure, and environmental fluctuations. Our main result implies that the “community increases when rare” persistence criterion also applies to these models. Our model, assumptions, and a definition of stochastic persistence are presented in Section 2. Except for a compactness assumption, our assumptions are quite minimal allowing for overcompensating density dependence and correlated environmental fluctuations. Long-term growth rates for these models and our main theorem are stated in Section 3. We apply our results to stochastic models of predator-prey interactions, stage-structured beetle dynamics, and competition in spatial heterogenous environments. The stochastic models for predator-prey interactions are presented in Section 4 and examine to what extent “colored” environmental fluctuations facilitate predator-prey coexistence. In Section 5, we develop precise criteria for persistence and exclusion for structured single species models and apply these results to the classic stochastic model of larvae-pupae-adult dynamics of flour beetles (Costantino et al., 1995; Dennis et al., 1995; Costantino et al., 1997; Henson and Cushing, 1997) and metapopulation dynamics (Harrison and Quinn, 1989; Gyllenberg et al., 1996; Metz and Gyllenberg, 2001; Roy et al., 2005; Hastings and Botsford, 2006; Schreiber, 2010). We show, contrary to initial expectations, that multiplicative noise with logarithmic means of zero can facilitate persistence. In Section 6, we examine spatial-explicit lottery models (Chesson, 1985, 2000a, 2000b) to illustrate how spatial and temporal heterogeneity, collectively, mediate coexistence for transitive and intransitive competitive communities. Proofs of most results are presented in Section 8.

## 2. Model and assumptions

We study the dynamics of interacting populations in a random environment. Each individual in population can be in one of individual states such as their age, size, or location. Let denote the row vector of populations abundances of individuals in different states for population at time . lies in the non-negative cone . The population state is the row vector that lies in the non-negative cone where . To account for environment fluctuations, we consider a sequence of random variables, where represents the state of the environment at time .

To define the population dynamics, we consider projection matrices for each population that depend on the population state and the environmental state. More precisely, for each , let be a non-negative, matrix whose -th entry corresponds to the contribution of individuals in state to individuals in state e.g. individuals transitioning from state to state or the mean number of offspring in state produced by individuals in state . Using these projection matrices and the sequence of environmental states, the population dynamic of population is given by

 Xit+1=XitAi(ξt+1,Xt).

where multiplies on the left hand side of as it is a row vector. If we define to be the block diagonal matrix , then the dynamics of the interacting populations are given by

 (1) Xt+1=XtA(ξt+1,Xt).

For these dynamics, we make the following assumptions:

1. is an ergodic stationary sequence in a compact Polish space (i.e. compact, separable and completely metrizable).

2. For each , is a continuous map into the space of non-negative matrices.

3. For each population , the matrix has fixed sign structure corresponding to a primitive matrix. More precisely, for each , there is a , non-negative, primitive matrix such that the --th entry of equals zero if and only if -th entry equals zero for all and .

4. There exists a compact set such that for all , for all sufficiently large.

Our analysis focuses on whether the interacting populations tend, in an appropriate stochastic sense, to be bounded away from extinction. Extinction of one or more population corresponds to the population state lying in the extinction set

 S0={x∈S:∏i∥xi∥=0}

where corresponds to the –norm of . Given , we define stochastic persistence in terms of the empirical measure

 (2) Πxt=1tt∑s=1δXs

where denotes a Dirac measure at , i.e. if and otherwise for any Borel set . These empirical measures are random measures describing the distribution of the observed population dynamics up to time . In particular, for any Borel set ,

 Πxt(B)=#{1≤s≤t|Xs∈B}t

is the fraction of time that the populations spent in the set . For instance, if we define

 Sη={x∈S:∥xi∥≤η for some i},

then is the fraction of time that the total abundance of some population is less than given .

###### Definition 2.1.

The model (1) is stochastically persistent if for all , there exists such that, with probability one,

 Πxt(Sη)≤ε

for sufficiently large and .

The set corresponds to community states where one or more populations have a density less than . Therefore, stochastic persistence corresponds to all populations spending an arbitrarily small fraction of time at arbitrarily low densities.

## 3. Results

### 3.1. Long-term growth rates and a persistence theorem

Understanding persistence often involves understanding what happens to each population when it is rare. To this end, we need to understand the propensity of the population to increase or decrease in the long term. Since

 Xit=Xi0Ai(ξ1,X0)Ai(ξ2,X1)…Ai(ξt,Xt−1),

one might be interested in the long-term “growth” of random product of matrices

 (3) Ai(ξ1,X0)Ai(ξ2,X1)…Ai(ξt,Xt−1)

as . One measurement of this long-term growth rate when is the random variable

 (4) ri(x)=limsupt→∞1tlog∥Ai(ξ1,X0)Ai(ξ2,X1)…Ai(ξt,Xt−1)∥.

Population is tending to show periods of increase when and asymptotically decreasing when . Since, in general, the sequence

 {1tlog∥Ai(ξ1,X0)Ai(ξ2,X1)…Ai(ξt,Xt−1)∥}∞t=1

does not converge, the instead of in the definition of is necessary. However, as we discuss in Section 3.2, the can be replaced by on sets of “full measure”.

An expected, yet useful property of is that with probability one whenever . In words, whenever population is present, its per-capita growth rate in the long-term is non-positive. This fact follows from being bounded above for . Furthermore, on the event of , we get that with probability one. In words, if population ’s density infinitely often is bounded below by some minimal density, then its long-term growth rate is zero as it is not tending to extinction and its densities are bounded from above. Both of these facts are consequences of results proved in the Appendix (i.e. Proposition 8.10, Corollary 8.17 and Proposition 8.19).

Our main result extends the persistence conditions discussed in the introduction to stochastic models of interacting, structured populations. Namely, if the community increases on average when rare, then the community persists. More formally, we prove the following theorem in the Appendix.

###### Theorem 3.1.

If there exist positive constants such that

 (5) ∑ipiri(x)>0 with probability one

for all , then the model (1) is stochastically persistent.

For two competing species () that persist in isolation (i.e. and with probability one), inequality (5) reduces to the classical mutual invasibility condition. To see why, consider a population state supporting species . Since species can persist in isolation, Proposition 8.19 implies that with probability one. Hence, inequality (5) for this initial condition becomes with probability one for all initial conditions supporting species . Similarly, inequality (5) for an initial condition supporting species becomes with probability one. In words, stochastic persistence occurs if both competitors have a positive per-capita growth rate when rare. A generalization of the mutual invasibility condition to higher dimensional communities is discussed at the end of the next subsection.

### 3.2. A refinement using invariant measures

The proof of Theorem 3.1 follows from a more general result that we now present. For this result, we show that one need not verify the persistence condition (5) for all in the extinction set . It suffices to verify the persistence condition for invariant measures of the process supported by the extinction set.

###### Definition 3.2.

A Borel probability measure on is an invariant measure for the model (1) provided that

1. for all Borel sets , and

2. if for all Borel sets , then for all Borel sets and .

Condition (i) ensures that invariant measure is consistent with the environmental dynamics. Condition (ii) implies that if the system initially follows the distribution of , then it follows this distribution for all time. When this occurs, we say is stationary with respect to . One can think of invariant measures as the stochastic analog of equilibria for deterministic dynamical systems; if the population statistics initially follow , then they follow for all time.

When an invariant measure is statistically indecomposable, it is ergodic. More precisely, is ergodic if it can not be written as a convex combination of two distinct invariant measures, i.e. if there exist and two invariant measures such that , then .

###### Definition 3.3.

If is stationary with respect to , the subadditive ergodic theorem implies that is well-defined with probability one. Moreover, we call the expected value

 ri(μ)=∫E[ri(X0)|X0=x,ξ1=ξ]μ(dξ,dx)

to be long-term growth rate of species with respect to . When is ergodic, the subadditive ergodic theorem implies that equals for -almost every .

With these definitions, we can rephrase Theorem 3.1 in terms of the long-term growth rates as well as provide an alternative characterization of the persistence condition.

###### Theorem 3.4.

If one of the following equivalent conditions hold

1. for every invariant probability measure with , or

2. there exist positive constants such that

 ∑ipiri(μ)>0

for every ergodic probability measure with , or

3. there exist positive constants such that

 ∑ipiri(x)>0 with probability one

for all

then the model (1) is stochastically persistent.

With Theorem 3.4’s formulation of the stochastic persistence criterion, we can introduce a generalization of the mutual invasibility condition to higher-dimensional communities. To state this condition, observe that for any ergodic, invariant measure , there is a unique set of species such that for all . In other words, supports the community . Proposition 8.19 implies that for all . Therefore, if is a strict subset of i.e. not all species are in the community , then coexistence condition (ii) of Theorem 3.4 requires that there exists a species such that . In other words, the coexistence condition requires that at least one missing species has a positive per-capita growth rate for any subcommunity represented by an ergodic invariant measure. While this weaker condition is sometimes sufficient to ensure coexistence (e.g. in the two species models that we examine), in general it is not as illustrated in Section 6.1. Determining, in general, when this “at least one missing species can invade” criterion is sufficient for stochastic persistence is an open problem.

## 4. Predator-prey dynamics in auto-correlated environments

To illustrate the applicability of Theorems 3.1 and 3.4, we apply the persistence criteria to stochastic models of predator-prey interactions, stage-structured populations with over-compensating density-dependence, and transitive and intransitive competition in spatially heterogeneous environments.

For unstructured populations, Theorem 3.4 extends Schreiber et al. (2011)’s criteria for persistence to temporally correlated environments. These temporal correlations can have substantial consequences for coexistence as we illustrate now for a stochastic model of predator-prey interactions. In the absence of the predator, assume the prey, with density at time , exhibits a noisy Beverton-Holt dynamic

 (6) Nt+1=Rt+1Nt1+aNt

where is a stationary, ergodic sequence of random variables corresponding to the intrinsic fitness of the prey at time , and corresponds to the strength of intraspecific competition. To ensure the persistence of the prey in the absence of the predator, assume and . Under these assumptions, Theorem 1 of Benaïm and Schreiber (2009) implies that converges in distribution to a positive random variable whenever . Moreover, the empirical measures with converge almost surely to the law of the random vector i.e. the probability measure satisfying for any Borel set .

Let be the density of predators at time and be the fraction of prey that “escape” predation during generation where is the predator attack rate. The mean number of predators offspring produced per consumed prey is , while corresponds to the fraction of predators that survive to the next time step. The predator-prey dynamics are

 (7) Nt+1 =Rt+1Nt1+aNtexp(−bPt) Pt+1 =cNt(1−exp(−bPt))+sPt.

To see that (7) is of the form of our models (1), we can expend the exponential term in the second equation. To ensures that (7) satisfies the assumptions of Theorem 3.1, we assume takes values in the half open interval . Since and , eventually enters and remains in the compact set

 S=[0,R∗/a]×[0,cR∗/(a(1−s))].

To apply Theorem 3.1, we need to evaluate for all with either or . Since converges to with probability one whenever , we have and whenever . Since with converges almost surely to , Proposition 8.19 implies . Moreover,

 (8) r2((N,0))=E[ln(cbˆN+s)]=∫ln(cbx+s)ν(dx)

By choosing and for sufficiently small (e.g. ), we have whenever if and only if

 (9) E[ln(cbˆN+s)]>0.

Namely, the predator and prey coexist whenever the predator can invade the prey-only system. Since is a concave function of the prey density and the predator life history parameters , Jensen’s inequality implies that fluctuations in any one of these quantities decreases the predator’s growth rate.

To see how temporal correlations influence whether the persistence criterion (9) holds or not, consider an environment that fluctuates randomly between good and bad years for the prey. On good years, takes on the value , while in bad years it takes on the value . Let the transitions between good and bad years be determined by a Markov chain where the probability of going from a bad year to a good year is and the probability of going from a good year to a bad year is . For simplicity, we assume that in which case half of the years are good and half of the years are bad in the long run. Under these assumptions, the persistence assumption for the prey is

To estimate the left-hand side of (9), we consider the limiting cases of strongly negatively correlated environments () and strongly positively correlated environments (). When , the environmental dynamics are nearly periodic switching nearly every other time step between good and bad years. Hence, one can approximate the stationary distribution by the positive, globally stable fixed point of

which is given by . Hence, if , then the distribution of approximately puts half of its weight on and half of its weight on and the persistence criterion (9) is approximately

Next, consider the case that in which there are long runs of good years and long runs of bad years. Due to these long runs, one expects that half time is near the value and half the time it is near the value . If , then the persistence criterion is approximately

Relatively straightforward algebraic manipulations (e.g. exponentiating the left hand sides of (10) and (11) and multiplying by ) show that the left hand side of (10) is always greater than the left hand side of (11).

###### Biological Interpretation 4.1.

Positive autocorrelations, by increasing variability in prey density, hinders predator establishment and, thereby, coexistence of the predator and prey. In contrast, negative auto-correlations by reducing variability in prey density can facilitate predator-prey coexistence (Fig. 1).

## 5. Application to structured single species models

For single species models with negative-density dependence, we can prove sufficient and necessary conditions for stochastic persistence. The following theorem implies that stochastic persistence occurs if the long-term growth rate when rare is positive and asymptotic extinction occurs with probability one if this long-term growth rate is negative.

###### Theorem 5.1.

Assume that (i.e. there is one species), H1-H4 hold and the entries of are non-increasing functions of . If , then

 (12) Xt+1=XtA(ξt+1,Xt)

is stochastically persistent. If , then with probability one.

Our assumption that the entries are non-increasing functions of ensures that for all which is the key fact used in the proof of Theorem 5.1. It remains an open problem to identify other conditions on that ensure for all .

###### Proof.

The first statement of this theorem follows from Theorem 3.1.

Assume that . Provided that is nonnegative with at least one strictly positive entry, Ruelle’s stochastic version of the Perron Frobenius Theorem (Ruelle, 1979a, Proposition 3.2) and the entries of being non-increasing in imply

 limt→∞1tlog∥Xt∥≤limt→∞1tlog∥X0A(ξt,0)…A(ξ1,0)∥=r(0)<0

with probability one. Hence, with probability one. ∎

Theorem 5.1 extends Theorem 1 of Benaïm and Schreiber (2009) as it allows for over-compensating density dependence and makes no assumptions about differentiability of . To illustrate its utility, we apply this result to the larvae-pupue-adult model of flour beetles and a metapopulation model.

### 5.1. A stochastic Larvae-Pupae-Adult model for flour beatles

An important, empirically validated model in ecology is the “Larvae-Pupae-Adult” (LPA) model which describes flour beetle population dynamics (Costantino et al., 1995; Dennis et al., 1995; Costantino et al., 1997). The model keeps track of the densities of larvae, pupae, and adults at time . Adults produce eggs each time step. These eggs are cannibalized by adults and larvae at rates and , respectively. The eggs escaping cannibalism become larvae. A fraction of larvae die at each time step. Larvae escaping mortality become pupae. Pupae are cannibalized by adults at a rate . Those individuals escaping cannibalism become adults. A fraction of adults survive through a time step. These assumptions result in a system of three difference equations

 (13) ℓt+1 =batexp(−celℓt−ceaat) pt+1 =(1−μl)ℓt at+1 =(ptexp(−cpaat)+(1−μa)at)

Environmental fluctuations have been included in these models in at least two ways. Dennis et al. (1995) assumed that each stage experienced random fluctuations due to multiplicative factors such that for are independent and normally distributed with mean zero i.e. on the log-scale the average effect of environmental fluctuations are accounted for by the deterministic model. Alternatively, Henson and Cushing (1997) considered periodic fluctuations in cannibalism rates due to fluctuations in the size of the habitat i.e. the volume of the flour. In particular, they assumed that for , for positive constants . If we include both of these stochastic effects into the deterministic model, we arrive at the following system of random difference equations.

 (14) ℓt+1 =batexp(−κelℓt/Vt+1−κeaat/Vt+1+ξlt) pt+1 =(1−μl)ℓtexp(ξpt) at+1 =(ptexp(−κpaat/Vt+1)+(1−μa)at)exp(ξat)

We can use Theorem 3.4 to prove the following persistence result. In the case of with probability one for , this theorem can be viewed as a stochastic extension of Theorem 4 of Henson and Cushing (1997) for periodic environments.

###### Theorem 5.2.

Assume for , for , ,,, and are ergodic and stationary sequences such that for , and some , and for some with probability one. Then there exists a critical birth rate such that

Extinction:

If , then converges almost surely to as .

Stochastic persistence:

If , then the LPA model is stochastically persistent.

Moreover, if with probability one and , then .

Remark. The assumption that are compactly supported formally excludes the normal distributions used by Dennis et al. (1995). However, truncated normals with a very large can approximate the normal distribution arbitrarily well. The assumption for some is more restrictive. However, from a biological standpoint, it is necessary as this term corresponds to the fraction of adults surviving to the next time step. None the less, we conjecture that the conclusions of Theorem 5.2 hold when are normally distributed with mean .

Theorem 5.2 implies that including multiplicative noise with -mean zero has no effect on the deterministic persistence criterion when with probability one. However, when these random variables are not perfectly correlated, we conjecture that this form of multiplicative noise always decreases the critical birth rate (Fig. 2). To provide some mathematical evidence for this conjecture, we compute a small noise approximation for the per-capita growth rate when the population is rare (Ruelle, 1979a; Tuljapurkar, 1990). Let

 Bt= ⎛⎜ ⎜⎝0(1−μl)exp(ξpt)000exp(ξat)bexp(ξlt)0(1−μa)exp(ξat)⎞⎟ ⎟⎠

be the linearization of the stochastic LPA model (14) at . Assume that where and . Ruelle (1979a, Theorem 3.1) implies that is an analytic function of . Therefore, one can perform a Taylor’s series expansion of as function of about the point . As we shall shortly show, the first non-zero term of this expansion is of second order. Expanding to second order in yields

 Bt≈ ⎛⎜⎝0(1−μl)0001b0(1−μa)⎞⎟⎠=B(I+εdiag{Zlt,Zpt,Zat}+ε2diag{Zlt,Zpt,Zat}2/2).

The entries of the second order term are positive due to the convexity of the exponential function. Hence, Jensen’s inequality implies that fluctuations in increase the mean matrix . This observation, in and of itself, suggests that fluctuations in increase . However, to rigorously verify this assertion, let and be the left and right Perron-eigenvectors of such that and . Let be the associated Perron eigenvalue of . Provided the are independent in time, a small noise approximation for the stochastic growth rate of the random products of is

 (15) r∗(δ0)≈logr0+ε22⎛⎝E[∑iviwi(Zit)2]−E⎡⎣(∑iviwiZit)2⎤⎦⎞⎠

Since the function is strictly convex and is a convex combination of , and , Jensen’s inequality implies

 (∑iviwiZit)2≤∑iviwi(Zit)2.

Therefore

It follows that the order correction term in (15) is non-negative and equals zero if and only if with probability one. Therefore, “small” multiplicative noise (with -mean zero) which isn’t perfectly correlated across the stages increases the stochastic growth rate and, therefore, decreases the critical birth rate required for stochastic persistence.

###### Biological Interpretation 5.3.

For the LPA model, there is a critical mean fecundity, above which the population persists and below which the population goes asymptotically to extinction. Fluctuations in the log survival rates decrease the critical mean fecundity unless the log survival rates are perfectly correlated.

###### Proof of Theorem 5.2.

We begin by verifying H1H4. H1 and H2 follow from our assumptions. To verify H3, notice that the sign structure of the nonlinear projection matrix for (14) is given by

 C=⎛⎜⎝010001101⎞⎟⎠

Since

 C4=⎛⎜⎝111112213⎞⎟⎠

has the sign structure of the primitive matrix for all and . Finally, to verify H4, define

 K=be2M−1/κea

Then

 ℓt+1≤batexp(−κeaat/Vt+1+ξlt)≤batexp(−κeaatexp(−M)+M)≤K

for all . Therefore, for and

 pt≤ℓt−1eM≤KeM

for all . Hence,

 at+1≤pteM+(1−δ)at

for all which implies for sufficiently large. The compact forward invariant set satisfies H4.

At low density we get

 Bt=A(ξt,0)=⎛⎜ ⎜⎝0(1−μl)exp(ξpt)000exp(ξat)bexp(ξlt)0(1−μa)exp(ξat)⎞⎟ ⎟⎠

Define to be the dominant Lyapunov exponent of the random products of . Note that with the notation of Theorem 3.4, . Theorem 3.1 of Ruelle (1979a) implies that is differentiable for and the derivative is given by (see, e.g., section 4.1 of Ruelle (1979a))

 r′(b)=E[vt(b)E31wt+1(b)vt(b)Bt(b)wt+1(b)]>0

where are the normalized left and right invariant sub-bundles associated with and is the matrix with in the entry and entries otherwise. Since the numerator and denominators in the expectation are always positive, is a strictly increasing function of . Since and , there exists such that for and for .

If , then and Theorem 5.1 implies that (14) is stochastically persistent. On the other hand, if , then and Theorem 5.1 implies that converges to with probability one as .

The final assertion about the stochastic LPA model follows from observing that if with probability one for all , then

 Bt=⎛⎜⎝0(1−μl)0001b0(1−μa)⎞⎟⎠exp(ξlt)

with probability one. Hence, where is the dominant eigenvalue of the deterministic matrix

 ⎛⎜⎝0(1−μl)0001b0(1−μa)⎞⎟⎠

Therefore, if , then . Using the Jury conditions, Henson and Cushing (1997) showed that if and if . Hence, when with probability one and , equals as claimed. ∎

### 5.2. Metapopulation dynamics

Interactions between movement and spatio-temporal heterogeneities determine how quickly a population grows or declines. Understanding the precise nature of these interactive effects is a central issue in population biology receiving increasing attention from theoretical, empirical, and applied perspectives (Petchey et al., 1997; Lundberg et al., 2000; Gonzalez and Holt, 2002; Schmidt, 2004; Roy et al., 2005; Boyce et al., 2006; Hastings and Botsford, 2006; Matthews and Gonzalez, 2007; Schreiber, 2010).

A basic model accounting for these interactions considers a population living in an environment with patches. Let be the number of individuals in patch at time . Assuming Ricker density-dependent feedbacks at the patch scale, the fitness of an individual in patch is at time , where is the maximal fitness and measures the strength of infraspecific competition. Let be the fraction of the population from patch that disperse to patch . Under these assumptions, the population dynamics are given by

 (16) Xrt+1=n∑s=1dsrλstXstexp(−αsXst)r=1,…,n.

To write this model more compactly, let be the diagonal matrix with diagonal entries , and be the matrix whose -th entry is given by . With this notation, (16) simplifies to

 Xt+1=XtF(Xt,λt)D

If are ergodic and stationary, take values in a positive compact interval and is a primitive matrix, then the hypotheses of Theorem 5.1 hold. In particular, stochastic persistence occurs only if , corresponding to the dominant Lyapunov exponent of the random matrix product , is positive.

When populations are fully mixing (i.e. for all ), Metz et al. (1983) derived a simple expression for given by

 (17)