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Extinction Dynamics of Age-Structured Populations in A Proc. Nadl. Acad. Sci. USA Vol. 85, pp. 7418-7421, October 1988 Population Biology Extinction dynamics of age-structured populations in a fluctuating environment (demography/ecology/life history/stochastic model/diffusion process) RUSSELL LANDE AND STEVEN HECHT ORZACK Department of Ecology and Evolution, University of Chicago, Chicago, IL 60637 Communicated by David M. Raup, June 3, 1988 (received for review February 2, 1988) ABSTRACT We model density-independent growth of an a recursion formula for the numbers of individuals in each age- (or stage-) structured population, assuming that mortality age class or developmental stage at time t + 1 in terms of and reproductive rates fluctuate as stationary time series. An- those at the previous time t, alytical formulas are derived for the distribution of time to extinction and the cumulative probability of extinction before n(t + 1) = A(t)n(t), [1] a certain time, which are determined by the initial age distri- bution, and by the infinitesimal mean and variance, # and a2, where n(t) is a column vector with entries representing the of a diffusion approximation for the logarithm of total popula- numbers of (female) individuals of each type at time t and tion size. These parameters can be estimated from the average A(t) is a square projection matrix with nonnegative entries life history and the pattern of environmental fluctuations in Aij(t) that determine the number of individuals of type i pro- the vital rates. We also show that the distribution of time to duced at time t + 1 per individual of type j at time t. For an extinction (conditional on the event) depends on the magnitude age-structured population, A(t) is the familiar Leslie matrix but not the sign of t. When the environmental fluctuations in with age-specific fecundity rates in the top row, age-specific vital rates are small or moderate, the diffusion approximation survival probabilities along the subdiagonal, and zeros else- gives accurate estimates of cumulative extinction probabilities where (12, 13). We assume that the projection matrices obtained from computer simulations. change with time such that each of the entries, or vital rates, compose a stationary time series. The theory of extinction times for single populations has ap- Under a mild assumption on the life history (nonperiodic plications in diverse fields, including paleontology, island fertilities), regardless of the initial population vector n(0), the biogeography, community ecology, and conservation biolo- probability distribution of the natural logarithm of total pop- gy. Various analytical models have been constructed to esti- ulation size asymptotically approaches a normal distribution mate extinction times, probabilities of extinction, or both for (2, 10, 14). For most life histories the rate of approach to a population subject to stochastic variation in demographic normality will be sufficiently fast that even for short times parameters (e.g., refs. 1-7). Two sources of random varia- the changes in logarithmic population size can be accurately tion in population growth can be distinguished: first, demo- approximated as a diffusion process with constant infinitesi- graphic stochasticity caused by finite population size, in mal mean Au and infinitesimal variance cr2 (10). which each individual independently experiences age-specif- Initial Age Distribution. The initial age distribution can ic probabilities of survival and reproduction; and second, en- have a substantial effect on extinction probabilities because vironmental stochasticity, which affects the vital rates of all individuals of different ages contribute unequally to future individuals similarly. Demographic stochasticity is impor- population growth. The influence of the initial age distribu- tant only in small populations, since chance variation in vital tion can be approximated in the following manner. Consider rates among individuals tends to average out in large popula- a sample path of the stochastic process from time 0 to time t, tions (greater than about 100 individuals), whereas environ- expressed as a product of t projection matrices (Eq. 1). Stan- mental factors can produce substantial fluctuations in demo- dard demographic reasoning applied to the stochastic case graphic parameters in populations of any size. In natural (e.g., ref. 15, page 51) indicates that asymptotically populations subject to substantial abiotic and biotic pertur- bations, environmental stochasticity is usually far more im- n(t) ==: R(O, t)(v(O) - n(O))u(t), [2] portant than demographic stochasticity in causing extinction (6, 7). Previous analytical models of stochastic population where R(0, t) is a scalar representing the growth of the ma- growth and extinction either lacked age structure or dealt trix product and v(0) and u(t) are the dominant left and right only with demographic stochasticity (1-7). eigenvectors of the matrix product, normalized so that Here we analyze the extinction dynamics of a population v(0) u(t) = 1. Both v(0) and u(t) geometrically converge to subject to environmental fluctuations in age-specific birth stationary distributions that are independent of t. In addi- and death rates. The theory of population growth developed tion, it is possible to show by using the methods of Tuljapur- by Cohen (8, 9) and Tuljapurkar and Orzack (10, 11) is used kar and Lee (personal communication), that v(0) = v + to derive a diffusion approximation for the logarithm of total O(E2), where v is the dominant left eigenvector of the aver- population size in a population subject to density-indepen- age projection matrix and E measures the size of the devi- dent fluctuations in vital rates. ations of the vital rates from the average matrix. Hence, for small noise, assuming u(t) is normalized so that 5iui(t) = 1, THE MODEL the initial total reproductive value in the population, V0 = Assumptions. A standard model of density-independent v n(0), gives an accurate approximation of the effect of ini- growth in an age- (or stage-) structured population specifies tial age distribution on the total population size at time t. Infinitesimal Mean and Variance. For "small" fluctuations The publication costs of this article were defrayed in part by page charge in the vital rates, the analytical framework in Tuljapurkar payment. This article must therefore be hereby marked "advertisement" (11) can be used to derive approximations for the constant in accordance with 18 U.S.C. §1734 solely to indicate this fact. asymptotic rates of change in mean and variance of the prob- 7418 Downloaded by guest on October 2, 2021 Population Biology: Lande and Orzack Proc. NatL Acad Sci. USA 85 (1988) 7419 ability distribution of the logarithm of population size. We with the initial condition p(x, Olxo) = 8(x - xO), the Dirac present for simplicity only the case of serially independent delta function located at xO. In the absence of an extinction environments, although his general formulas include terms boundary (absorbing barrier) Eq. 7 has natural boundaries at describing serial correlation of fluctuations in the vital rates. -00 and +00 and the solution is a normal distribution with The infinitesimal mean, g, also known as the long-run mean xO + ,a and variance cat (18), as described by the as- growth rate of the population, and the infinitesimal variance, ymptotic results given above. a.2, are More realistically, the imposition of an extinction bound- ary-i.e., certain extinction below a population size of one k -In Ao -a2/2 [3] individual, dictates the boundary condition and P(O, tlxo) = 0. [8] Quantities pertaining to a population declining to any thresh- old can be obtained by taking xO to be the distance from the where AO is the dominant eigenvalue of the average projec- adjusted initial size to the threshold on the log scale (3). Pre- tion matrix A (containing the average vital rates Aij) and 8 is viously known solutions for the Wiener process with an ab- a column vector (and ST its transpose) of sensitivity coeffi- sorbing barrier (18) merely require a linear transformation of cients that can be expressed as aAo/aAij = viau, in which vi coordinates to apply to the present problem (Eqs. 9-12). The and uj are elements of the dominant left and right eigenvec- solution to Eqs. 7 and 8 is tors of A, normalized as above (16). C is the variance-covari- ance matrix of fluctuations in the elements of A(t). (,u here is p(x, tlxo) = (21ror2t)-"2[exp-(x - xO - Lt)2/2cr2t} equivalent to a in refs. 10 and 11 and to In A in refs. 8,9 , and 14). -exp{-2pLxo/o2 - (x + Xo - ,ut)2/2a'2t}]. [9] Another way of obtaining estimates of the infinitesimal mean and variance in the diffusion approximation for loga- The probability ofextinction in the interval t to t + dt, denot- rithm of population size is by numerical simulation of the ed as g(tlxo)dt, is derived from the rate of decrease of the stochastic process. The total population size at time t obeys total probability that the population exists at time t, the recursion equation N(t + 1) = A(t)N(t) where the values d x of the realized geometric growth rate, A(t), are determined g(tlxo) = -(dldt)J p(x, tlxo)dx [1Oal by the projection matrix for the age-structured population (Eq. 1). Defining x(t) = In N(t) and r(t) = In A(t), the recur- = X - + At)2/2a2t}. sion for the natural logarithm of total population size, x(t + xO(2 cr2t3)f-1/2exp{ (xO [lOb] 1) = r(t) + x(t), has the solution for a particular sample path The cumulative probability that the population becomes ex- x(t) = x(O) + >Lflr(i). The r(i) have serial correlation caused tinct before time t is, from Eqs. 9 and 1Oa, by the time lag or "momentum" inherent in age-structured populations, in addition to possible environmentally induced autocorrelation in the vital rates.
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