BEHAVIOR OF CAREY’S EQUALITY IN TWO-DIMENSIONS: AGE AND PROPORTION OF POPULATION Arni S.R. Srinivasa Rao Augusta University 1120 15th Street Augusta, GA 30912, USA Email: arrao@augusta..edu James R. Carey Department of Entomology University of California, Davis, CA 95616 USA and Center for the Economics and Demography of Aging University of California, Berkeley, CA 94720 Email: jrcarey@ucdavis.edu Abstract. Carey’s Equality arises while studying the average time left to die after forming a captive cohort of individuals in a stationary population. This equality was first time observed in experimental entomology. We extend the ideas of Carey’s Equality, which arises in stationary populations to two dimensions with the aim to answer questions related to aging, age-structure of the captive cohort. These ideas found to be useful for identifying individuals by age group in a captive cohort. Key words: Captive cohort, population age-structure, partition function. MSC: 92D25 Contents 1. 2 Introduction 2. 3 Structure of the 2-Dimensional Captive Cohort 3. 5 Captive population Age-structure, Truncation and Partition Functions 3.1. Time left for a captive cohort and right truncation 6 4. Discussion 11 5. Conclusions 11 References 11 1
BEHAVIOR OF CAREY’S EQUALITY IN TWO-DIMENSIONS: AGE AND PROPORTION OF POPULATION 2 1. Introduction Although one of the basic canons of population studies is that knowledge of age composition is fundamental to the understanding the dynamics of a population, information on the age structure of a natural population is usually extremely limited. Although there are some long-term animals studies in which most or even all newborn are marked at birth and thus are of known age throughout a study[14], investigations with this information are the exception rather than the rule because of costs and logistics. Similar challenges exist for studies of natural arthropod populations where various methods are used to estimate the age of individuals including, for example, mechanical damage, biochemical assays, physiological markers, and gene expression[15]. Because of the disconnect between the need for information on population age structure but the technical and budgetary challenges of obtaining the age of individuals for estimating age structure, James R. Carey and his colleagues developed a new approach to obtaining information about the age structure of wild populations that involved neither mark-recapture or age-estimating technologies. Rather the method was based on the concept that the death patterns observed in captured or marked individuals of unknown age would yield information on the age structure of the population from which these individuals were sampled. The starting point for this concept was to consider a closed stationary (zero growth) population in which the pre- and post-capture actuarial environments were identical and from which individuals were sampled at random and monitored through death. Using a simple table (See Table 1 in [6]) Carey showed that, under the stationarity assumption, that the post-capture death distribution of the randomly sampled individuals equals the age structure of the population. In this same paper Müller and his colleagues expressed this relationship analytically. See [1, 2] for a historical perspective on Carey’s Equality and related new developments. When population is stable, then the age-structure of the population is sensitive to the momentum [5]. Carey’s Equality[3] and a generalized theorem on Carey’s Equality[4] are true when the population growth is stationary. This equality is obtained from the information on age of an individual and life left at the time of capture of individual while forming a captive cohort in various experiments in ecology and population biology (for example, see [6, 7]). In one dimensional Carey’s Equality, f 1 ( a ) = f 2 ( a ) is true for any arbitrary age a of a population, where f 1 and f 2 are probability density functions of age composition and distribution of remaining lifespans in a stationary population [3]. This result is fundamental in establishing the experimental observations by Carey [6, 7]. f 1 ( a ) = f 2 ( a ) and other results describing the symmetries of life lived and remaining lifespans in [8], can be obtained as a direct consequences and applications of classical renewal theory frameworks [9, 10, 11, 12]. A generalized theorem of one dimensional Carey’s Equality was proved for sufficiently large stationary population without standard life table and renewal theory framework [4]. It was shown, in this generalized framework, that, there exists a graph from the family of exhaustive graphs constructed based on the co-ordinates of ages of individuals and their corresponding lives lived, will be equivalent to the graph obtained by the sequence of co-ordinates of the ages and corresponding lengths of life remaining (or life left) for each individual in a captive cohort [4]. All these above results are helpful in explaining the pattern of remaining lifespans for certain captive cohorts observed by Carey. Less
BEHAVIOR OF CAREY’S EQUALITY IN TWO-DIMENSIONS: AGE AND PROPORTION OF POPULATION 3 is known on determining an age structure of a captive cohort which is unknown at the time of formation, which will otherwise be helpful in understanding age wise or age-group wise remaining lifespans. Understanding age structure in a captive cohort formed for ecological and biodemographic experiments could be very helpful in epidemiology studies for understanding transmission dynamics of parasites in some vector borne diseases. The purpose of the current paper is to introduce a variation on Carey’s Equality in which the post-capture period of monitoring is truncated to less than the maximal age of individuals sampled in the hypothetical population. We introduce and develop concepts of two dimensional Carey’s Equality by considering age structure and proportion of population continuous in a range of entire age interval. We use these concepts for constructing partial age structure of the captured population which is unknown. We also consider truncation from right during the follow-up of captive cohort as an additional variable and use this variable in the partition function constructed in the process. 2. Structure of the 2-Dimensional Captive Cohort Let x ( a, i ) be the i th − individual who was captured at age a and y ( a, i ) be the life left for i th − individual who was captured at age a. Then we represent the total number of individuals captured ´ ∞ ´ ∞ ´ ∞ at age a as 0 x ( a, i ) di, who will have life left as 0 y ( a, i ) di units. Due to aging, 0 x ( a, i ) di ´ ∞ will become 0 x ( b, j ) dj during the time period b − a ( b > a ). Suppose on the positive quadrant, we ´ ∞ ´ ∞ � � � � plot A, B, Q , and P where A = ( a, 0) , B = ( b, 0) , Q = 0 x ( b, j ) dj and P = 0 x ( a, i ) di , b, a, respectively (see Figure 2.1), then the area of the quadrilateral, ABQ ′ P, say, K, for BQ ′ > BQ , satisfies the inequality (2.1), � ˆ ∞ ˆ ∞ 1 � (2.1) K > 2 ( b − a ) x ( a, i ) di + x ( b, j ) dj 0 0 and the death rate per b − a time units is, � ˆ ∞ ˆ ∞ 1 � (2.2) x ( a, i ) di − x ( b, j ) dj ( b − a ) 0 0 If death rate is zero, then K becomes, ˆ ∞ (2.3) K = ( b − a ) x ( b, j ) dj 0 In the event that a captive cohort is formed from a stationary population at time t 0 , then ´ ∞ ´ ∞ ´ ∞ 0 x ( a, i, t 0 ) di is a part or a sub-cohort of the captive cohort whose size is 0 x ( s, i, t 0 ) dids 0 ´ ∞ ´ ∞ (here, s is a variable representing age). The sub-cohort 0 x ( a, i ) di will live for a 0 x ( a, i, t 0 ) di time units if for each individual one dimensional Carey’s Equality explained in [3, 4] is true. Then the relation between each age in a sub-cohort of a captive cohort, size of the sub-cohort and life lived by individuals in the sub-cohort is given in
BEHAVIOR OF CAREY’S EQUALITY IN TWO-DIMENSIONS: AGE AND PROPORTION OF POPULATION 4 Figure 2.1. Aging process in a captive cohort ´ ∞ 0 y ( a, i ) di (2.4) a = ´ ∞ 0 x ( a, i ) di The time left for a captive cohort describes the collective time that all the sub-cohorts of individuals ´ ∞ ´ ∞ ´ ∞ ´ ∞ of each age would live in the future. The time left for 0 x ( s, i, t 0 ) dids is 0 y ( s, i ) dids , 0 0 and if a captive cohort consists of individuals of only one age, a, then the total time left for the captive cohort is also a. We define a function of age-structure in the captive cohort, H 1 , and an associated set, A, such that, (2.5) : (0 , ∞ ) → J ( k ) H 1 where J ( k ) = { 1 , 2 , · · · , k } for sufficiently large k , and, (2.6) = { ( a, H 1 ( a )) ∀ a ∈ (0 , ∞ ) } A ´ ∞ where H 1 ( a ) = 0 x ( a, i ) di ∈ J ( k ) . If H 1 is constant, then, the captive cohort is uniformly distributed over all ages or captive cohort consists of individuals of same age. If H 1 is not constant, then, there are three possibilities: (i) H 1 ( a ) > H 1 ( a + h ) for all h > 0 , (ii) H 1 ( a ) < H 1 ( a + h ) for all h > 0 , (iii) H 1 is neither increasing nor decreasing for 0 < a < ∞ . We define a function of life left in a captive cohort, H 2 , as follows: (2.7) H 2 : ( t 1 , t 2 , · · · , t ∞ ) → J ( k ) for
Recommend
More recommend
