Population Stability and Momentum

POPULATION STABILITY AND MOMENTUM Arni S.R. Srinivasa Rao1,2,3 arXiv:1503.03373v1 [q-bio.QM] 10 Feb 2015 (Appeared in Notices of the American Mathematical Society) Abstract. A new approach is developed to understand stability of a population and further understanding of population momentum. These ideas can be generalized to populations in ecology and biology. 1. Introduction One commonly prescribed approach for understanding the stability of system of dependent variables is that of Lyapunov. In a possible alternative approach - when variables in the system have momentum then that can trigger additional dynamics within the system causing the system to become unstable. In this study stability of population is defined in terms of elements in the set of births and elements in the set of deaths. Even though the cardinality of the former set has become equal to the cardinality of the latter set, the momentum with which this equality has occurred determines the status of the population to remain at stable. Such arguments also works for the other population ecology problems. 2. Population Stability Theory Suppose |PN (t0 )| be the cardinality of the set of people, PN (t0 ), representing population at global level at time t0 , where PN (t0 ) = {u1 , u2, · · · , uN }, the elements u1 , u2 , · · · , uN represent individuals in the population. Broadly speaking, the Lyapunov stability principles (see [VLL]) suggests, |PN (t0 )| is asymptotically stable at population size U, if ||PN (T )| − U| < ǫ (ǫ > 0) at all T whenever T > t0 . In some sense, |PN (t0 )| attains the value U over the period of time. Lotka-Voltera’s predator and prey population models provide one of the classical and earliest stability analyses of population biology (see for example, [JDM]) and Lyapunov stability principles often assist in the analysis of such models. These models have equations that describe the dynamics of at least two interacting populations with parameters describing interactions and natural growth. Outside human population models and ecology models, stability also plays a very important role in understanding epidemic spread [AR]. In this paper, we are interested in factors that cause dynamics in PN and relate these factors with status of stability. A set of people PM (t0 ) = {um1 , um2 , · · · , umM }, where PM (t0 ) ⊂ PN (t0 ), are responsible for increasing the population (reproduction) during the period (t0 , s) and contribute to PN (s), the set of people at s (if they survive until the time s). The  set QM1 (s − t0 ) = vM11 , vM12 , · · · , vM1M1 represent removals (due to deaths) from PN (t0 ) during the time interval (t0 , s). Let Rφ (s − t0 ) be the period reproductive rate (net) applied on PM (t0 ) for the period (t0 , s), then the number of new population added during (t0 , s) is Rφ (s − t0 ) |PM (t0 )| . Net reproduction rate at time t0 (or in a year t0 ) is the average number of female children that 1 1. To commemorate MPE2013 launched by the International Mathematics Union, the author dedicates this work to Alfred J. Lotka. 2 Arni S.R. Srinivasa Rao is an Associate Professor at Georgia Regents University, Augusta, USA. His email address is [email protected] 3 Acknowledgements: Very useful comments by referees have helped to re-write some parts of the article for better readability. Comments from Professor JR Drake (University of Georgia, Athens), NV Joshi (Indian Institute of Science, Bangalore) and from Dr. Cynthia Harper (Oxford) helped very much in exposition of the paper. My sincere gratitude to all. Author is partially supported by DFID and BBSRC, UK in the form of CIDLID project BB/H009337 1 POPULATION STABILITY AND MOMENTUM 2 would be born to single women if she passes through age-specific fertility rates and age-specific mortality rates that are observed at t0 (for the year t0 ). Since net reproductive rates are futuristic measures, we use period (annual) reproductive rates for computing period (annual) increase in population. Let CN1 (s − t0 ) = {w1 , w2 , · · · , wN1 } be the set of newly added population during (t0 , s) to the set PN (t0 ). After allowing the dynamics during (t0 , s), the population at s will be PN (t0 ) ∪ CN1 (s − t0 ) − QM1 (s − t0 ) = (2.1) ( u : u ∈ PN (t0 ) ∪ CN1 (s − t0 ) and u ∈ / QM1 (s − t0 ) ) = {u1 , u2 , · · · , uN +N1 −M1 } Note that, QM1 (s−t0 ) ⊂ PN (t0 )∪CN1 (s−t0 ), because the set of elements {vM1 , vM12 , · · · , vM1M1 eliminated during the time period (t0 , s) are part of the set of elements {u1 , u2, · · · , uN , w1 , w2 , · · · , wN1 } and the resulting elements surviving by the time s are represented in equation (2.1). The element u1 in the set (2.1) may not be the same individual in the set PN (t0 ). Since we wanted to retain the notation that represents people living at each time point, so for ordering purpose, we have used the symbol u1 in the set (2.1). Using Cantor–Bernstein–Schroeder theorem [PS], |CN1 (s − t0 )| = |QM1 (s − t0 )| if |CN1 (s − t0 )| ≤ |QM1 (s − t0 )| and |CN1 (s − t0 )| ≥ |QM1 (s − t0 )| . If |CN1 (s − t0 )| = |QM1 (s − t0 )| then the natural growth of the population (in a closed situation) is zero and if this situation continues further over the time then the population could be termed as stationary. Assuming these two quantities are not same at t0 , the process of two quantities |CN1 | and |QM1 | becoming equal could eventually happen due to several sub-processes. Case I: |CN1 | > |QM1 | at time t0 . We are interested in studying the conditions for the process |CN1 | → |QM1 | for some s > t0 . Two factors play a major role in determining the speed of this process, they are, compositions of the family of sets [{PM (s)} : ∀s > t0 ] and [{Rφ (s)} : ∀s > t0 ]. Suppose |PM (s1 )| > |PM (s2 )| > · · · > |PM (sT )| but the family of {|Rφ (s)|} does not follow any decreasing pattern for some t0 < s1 < s2 < ... < sT < s, then |CN1 | 9 |QM1 | by the time sT . If Rφ (s1 ) > Rφ (s2 ) > ... > Rφ (sT ) for t0 < s1 < s2 < ... < sT < s such that |Rφ (sT − sT −1 ) |PM (sT −1 )| − QM1 (sT − sT −1 )| → 0 for some sufficiently large T > t0 and sufficiently small |Rφ (sT − sT −1 )|, then |CN1 | → |QM1 | by the time sT . Note that in an ideal demographic transition situation, both these quantities should decline over the period and the rate of decline of |QM1 | is slower than the rate of decline in |CN1 | because |CN1 | > |QM1 | at time t0 . Demographic transition theory, in simple terms, is all about, determinants, consequences and speed of declining of high rates of fertility and mortality to low levels of fertility and mortality rates. For introduction of this concept see [KD] and for an update of recent works, see [JC]. Above trend of |PM (s1 )| > |PM (s2 )| > · · · > |PM (sT )| (i.e. decline in people of reproductive ages over the time after t0 ) happens when births continuously decrease for several years. Following the trend Rφ (s1 ) > Rφ (s2 ) > ... > Rφ (sT ) will lead to decline in new born babies and this will indirectly result in decline in rate of growth of people who have reproductive potential. However the decline in |Rφ (s)| for s > t0 is well explained by social and biological factors, which need not follow any pre-determined mathematical model. However the trend in |Rφ (t)| for t < t0 can be explained using models by fitting parameters obtained from data. During the entire process the value of |QM1 | after time t0 is assumed to be dynamic and decreases further. If a population continues to remain at this stage of replacement we call it a stable population. The cycle of births, population aging and deaths is a continuous process with discretely quantifiable factors. Due to improvement in medical sciences there could be some delay in deaths, but eventually the aged population has to be moved out of {PN }, and consequently, population stability status can be broken with a continuous decline in {|CN1 |}. Case II: |CN1 | = |QM1 | at time t0 . It is important to ascertain whether this situation was immediately proceeded by case I or case II before determining the stability process. Suppose case POPULATION STABILITY AND MOMENTUM 3 Figure 2.1. (a) The cycle of all the cases could follow one after another and the quantity at which equality of CN1 and QM1 occurs determines the duration of the case II. (b) Some of the sub-populations which are not satisfied the equality of CN1 and QM1 is compensated by the other sub-populations which are satisfying either CN1 > QM1 or CN1 < QM1 . II is immediately preceded by case I, then the rapidity and magnitude at which the difference between |CN1 | and |QM1 | was shrunk prior to t0 need to be quantified. Let us understand the contributing factors for the set QM . At each t, there is a possibility that the elements from the sets CN1 , PN − CN1 − PM , PM are contributing to the set QM . Due to high infant mortality rates, the contribution of CN1 into QM is considered to be high, deaths of adults of reproductive ages, PM , and all other individuals (including the aged), PN − CN1 − PM , will be contributing to the set QM . Case II could occur when |CN1 | and |QM1 | are at higher values or at lower values. Equality at higher values possibly indicates, the number of deaths due to three factors mentioned here are high (including high old age deaths) and these are replaced by equal high number of births, i.e. |Rφ | and |PM | are usually high to reproduce a high birth numbers. If equality at lower values of |CN1 | and |QM1 | occurs after phase of case I then the chance of PN remaining in stable position is higher. Suppose elements of PN are arbitrarily divided into k−independent and non-empty subsets, A(1), ´k A(2), · · · ,A(k) such that |PN |= 1 |A(s)| ds. Let F be  the family of all the sets A(s) such that F be an arbitrary size of k ∗ of subset ∪ (A(s)) = PN . Members of F are disjoint. Suppose k∗   F of F are satisfying the case II and F − are not satisfying at time t0 and t > t0 , then we k∗ are not sure of total population also attains stability by Theorem 1. POPULATION STABILITY AND MOMENTUM  4  F Theorem 1. Suppose each of the member of is satisfying the condition |CN1 | = |QM1 | k∗   F are not satisfying the condition |CN1 | = |QM1 | at time t ≥ t0 , then this does not and F − k∗ always leads PN to stability. Proof. Note that F has collection of k−sets. Suppose a collection C divides CN1 into k−components ´k of subpopulations {CN1 (1), CN1 (2), · · · , CN1 (k)} such that |CN1 |= 1 |CN1 (s)| ds, where CN1 (s) is the sth − subset in C and a collection Q divides QM1 into k− components of subpopulations {QM1 (1), QM1 (2), · · · , QM1 (k)} such that ´k |QM1 | = 1 |CM1 (s)| ds, where QM1 (s) is the sth −subset in Q. By hypothesis, |CN1 (s∗ )| = |QM1 (s∗ )| for s∗ ∈ {1∗ , 2∗ , · · · , k ∗ } at each time t ≥ t0 until, say, tT . The order between k ∗ and k − k ∗ could be one of the following: 2k ∗ < k, 2k ∗ > k, k ∗ = k2 . Suppose CN1 ⊂ C and QM1 ⊂ Q with CN∗ 1 =  ∗ CN1 (1), CN∗ 1 (2), · · · , CN∗ 1 (k)  = Q∗M1 (1), Q∗M1 (2), · · · , Q∗M1 (k) QM1 for same above arbitrary combination of k ∗ −components and rest of the k − k ∗ components are ∗∗ ∗∗ satisfying CN∗∗1 (s∗∗ ) − Q∗∗ = 1, 2, · · · , k − k ∗ . We obtain unstable integral M1 (s ) 6= 0 for all s over all k − k ∗ components to ascertain the magnitude of unstability. ˆ (2.2) k−k ∗  ∗∗ ∗∗ CN∗∗1 (s∗∗ ) − Q∗∗ M1 (s ) ds  1 The stable integral for this situation is ˆ (2.3) k∗  1  CN∗ 1 (s∗ ) − Q∗M1 (s∗ ) ds∗ To check the unstable and stable points over the time period (t0 , tT ), one can compute following integrals: (2.4) ˆ tT t0 (2.5) ˆ k−k ∗ tT ˆ 1 ˆ t0   ∗∗ ∗∗ CN∗∗1 (s∗∗ ) − Q∗∗ M1 (s ) ds du   CN∗ 1 (s∗ ) − Q∗M1 (s∗ ) ds∗ du k∗ 1 ∗∗ For each of the k − k ∗ component, the values of CN∗∗1 (s∗∗ ) − Q∗∗ M1 (s ) can be either positive ∗∗ or negative. If at time t0 , for all s∗∗ = 1, 2, · · · , k − k ∗ , the values of CN∗∗1 (s∗∗ ) − Q∗∗ M1 (s ) are positive (or negative) then the eq. (2.2) will take a positive (or negative) quantity and the population at time t0 is not stable. If such a situation continues for all tT ≥ t0 , then the integral in eq. (2.4) would never become zero and the population remains unstable in the entire period ∗∗ (t0 , tT ). However, for some of the s∗∗ , if the quantity CN∗∗1 (s∗∗ ) − Q∗∗ M1 (s ) is positive and for ∗∗ other s∗∗ , if the quantity CN∗∗1 (s∗∗ ) − Q∗∗ M1 (s ) is negative such that eq. (2.2) is zero at each of the time points for the period (t0 , tT ) then the population remains stable during this period (because by hypothesis the eq. (2.5) is zero).  POPULATION STABILITY AND MOMENTUM 5 Case III. |CN1 | < |QM1 | at time t0 . Global occurrence of this case at lower values of |CN1 | and |QM1 | indicates that the PN is declining and also is in unstable mode. Rφ has been very low consistently for the period t < t0 and the supply to the set PM has diminished over a period in the past. All the subsets of CN1 and QM1 might not be stable in case III, but by similar arguments of the Theorem 1, global population behavior nullifies some of the local population and case III is still satisfied globally. All three cases would be repeated one following another. Most countries are currently facing case I with varying distance between |CN1 | and |QM1 | . 3. Replacement Metric We introduce a metric, dM , which we call a replacement metric, with a space, Mr as follows: Definition 2. (Replacement Metric). Let A1 = min {||CN1 (s)| − |QM1 (s)|| : s > t0 } and A2 = max {||CN1 (s)| − |QM1 (s)|| : s > t0 }. Let Mr = [A1 , A2 ] ⊂ R+ and M = {||CN1 (s)| − |QM1 (s)|| : s > t0 } with the metric dM (x, y) = |x−y| . We can verify that (M, dM ) is a metric space with 2 dM : (M × M) → Mr and non-empty set M. The metric M, in the definition 1 is bounded, because dM (x, y) < k for k > 0. Definition 3. Suppose ||CN1 (s1 )| − |QM1 (s1 )|| = f1 , ||CN1 (s2 )| −|QM2 (s2 )|| = f2 and so on for s1 < s2 < ... . Then we say population is stable if fsT → 0 for sufficiently large T and d |CN1 (sT )| = dsdT |CN1 (sT )| = 0. dsT 4. Conclusions We can prove that the value at which the population remains stable is variable, i.e. the value at which the population becomes unstable by deviating from case II could be different from the value (at a future point in time) population becomes stable when it converges to case II. Replacement metrics (see definition 2) are helpful in seeing this argument and such analysis is not possible by Lotka-Voltera or Lyupunov methods. Due to population momentum, there will be an increase in the population even though the reproduction rate of the population becomes below the replacement level. Population stability will always attain a local stable points before diverging and again converging at a local stable point. The duration of a local stable point depends on the density of the population and resources available for the population. References [VLL] V. Lakshmikantham, X.Z. Liu (1993). Stability analysis in terms of two measures. World Scientific Publishing Co., Inc., River Edge, NJ. [JDM] J.D. Murray (2003). Mathematical Biology I: An Introduction. Springer-Verlag. [AR] Arni S.R. Srinivasa Rao (2012). Understanding theoretically the impact of reporting of disease cases in epidemiology. J. Theoret. Biol. 302, 89–95. [PS] P. Suppes (1960). Axiomatic set theory. The University Series in Undergraduate Mathematics D. Van Nostrand Co., Inc. [KD] K. Davis (1945). The World Demographic Transition, Annals of the American Academy of Political and Social Science (237), pp. 1–11. [JC] J.C. Caldwell (2006). Demographic Transition Theory, (Edited Book). Springer, The Netherlands.