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  1. 1. The Ping-Pong Effect: Extinction Dynamics in Stochastic Populations with Migration Alexa Aucoin1 and Ryan Dykstra2 , Advisors: Dr. Eric Forgoston1 and Dr. Lora Billings1 1 Department of Mathematical Sciences and 2 Department of Chemistry and Biochemistry, Montclair State University, Montclair, NJ 07043 USA Abstract We consider a two-population stochastic model where both populations are connected through migration. Each population of species undergoes birth and annihilation events. Using a master equation ap- proach and a WKB (Wentzel-Kramers-Brillouin) approx- imation, we find the theoretical mean time to extinction. Numerical Monte Carlo simulations agree well with the analytical results, and also allow for an exploration of how changes in migration rates affect the behavior of extinc- tion events in both populations. Real World Implications Exploring extinction dynamics in stochastic, migratory pop- ulations, has an important public health implication. Under- standing what causes local and global disease die out makes infectious populations more manageable. Our research can be used to establish a control on infections in their endemic states and force them more quickly to extinction. Our Model We consider a simplified two-population model that under- goes birth and annihilation events [1]. The transition rates are Event W(X;r) X λ → 2X λX 2X σ → ∅ σX2 2 Y λ → 2Y λY 2Y σ → ∅ σY 2 2 X µ → Y µX Y µ → X µY . Let λ be the birth rate, σ the death rate and µ the migration rate between populations. We normalize X and Y by k1K, the local carrying capacity of population 1, and obtain the mean field equations ˙x = x − x2 − µx + µy ˙y = y − y2 κ + µx − µy, where κ = k2 k1 is a normalization factor. Theory and Analytic Work To analyze extinction dynamics, we use the model’s transitions rates, W(X;r), to formulate the master equation. The master equation provides the probability of finding the populations at size X and Y at time t, and is given by ˙ρ(X, t) = r [W(X − r; r)ρ (X − r, t) − W(X; r)ρ(X, t)] . (1) Here r is a uniform change in population and can be positive or negative. Due to the complexity of the master equation, it generally cannot be solved analytically. Since we are interested in extinction which is a rare event, we may approximate the master equation using a Wentzel- Kramers-Brillouin (WKB) ansatz P(x, t) = e−KS(x,t) . Here, S(x, t) is the action, and K is the carrying capacity [2]. From this we derive the effective Hamiltonian for our model. It is given by H = x(ePx − 1) + y(ePy − 1) + σ x2 2 (e−2Px − 1) + σ y2 2κ (e−2Py − 1) + µx(e−Px+Py − 1) + µy(ePx−Py − 1). (2) If we let µ = 0, the Hamiltonian becomes separable. Due to normalization, population Y is a scaled version of population X. We can look at one population to understand the behavior of both. The Hamiltonian for one population is H(x, px) = x(epx − 1) + σ x2 2 (e−2px − 1). (3) There are many paths to extinction. The optimal path, popt, describes the path to extinction that is most likely to occur. By setting Eq. (3) to zero and solving for px we obtain Sopt the action associated with the optimal path. Sopt = xext xend popt(x)dx (4) The limits here are the endemic and extinct states. The mean time to extinction, MTE, can now be calculated by τ = B exp  K 0 xend popt(x)dx  . (5) The pre-factor B accounts for a more accurate extinction time. For our model, the pre-factor is B = 2 √ πσ λ 3 2 . (6) Comparison with Numerics We use Monte Carlo simulations and compare with analytic results. In the figure to the right, the solid red line shows ln(τ) (Eq(5)), versus the birth rate, λ. The dashed red line is the theory shifted upwards by 0.15. The blue squares are numerical MTE found by 5000 realizations with σ = 1.5 and varied λ. The results show great agreement. 8 9 10 11 12 13 14 15 16 1.5 2 2.5 3 3.5 4 4.5 λ Ln(MTE) The Ping Pong Effect Let µ > 0. We numerically explore the effects of different values of µ and κ. Due to stochastic effects, it is possible for either population to go extinct. For example, population 1 may go extinct. However, this is a local extinction and may not be permanent. After some time, due to migration effects, population 2 can revive population 1. Now, there are two viable populations, each with the possibility to go extinct, and the process can repeat. We refer to this alternation of local extinctions as the Ping-Pong Effect. As µ increases, the MTE increases dramatically. For κ close to 1, we see many local extinctions in both popula- tions and a global extinction happening relatively quickly. On the other hand, for κ << 1, die out, or local extinction, is seen mostly in the smaller population and the time to global extinction is much longer. Universal parameters: λ = 1, K = 6 0 20 40 60 80 100 120 140 160 0 5 10 15 20 Time Population PopX PopY Figure 1: κ = .9, µ = .05, σ = .15 2000 2050 2100 2150 2200 2250 0 5 10 15 20 Time Population PopX PopY Figure 2: κ = .9, µ = .4, σ = .15 5000 5500 6000 6500 7000 0 10 20 30 40 Time Population PopX PopY Figure 3: κ = .5, µ = .05, σ = .0833 Future Work In the future, we will perform numerical analysis of the effect of migration on local and global extinction times. These studies will enable the establishment of a control to decrease the mean time to extinction. References [1]Michael Khasin, Baruch Meerson, Evgeniy Khain, and Leonard M. Sander Minimizing the Population Extinction Risk by Migration. Physical Review Letters E 109, 138104 (2012). [2] Michael Assaf and Baruch Meerson. Extinction of metastable stochastic populations. Physical Review E 81, 021116 (2010). This research was supported by the National Science Foun- dation Award # CMMI-1233397.