In probabilistic terms, the dynamics of a population with N individuals is no longer described by a deterministic functi

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In Probabilistic Terms The Dynamics Of A Population With N Individuals Is No Longer Described By A Deterministic Functi 1 (69.46 KiB) Viewed 35 times

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Related materials (Equation 8.2):

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In probabilistic terms, the dynamics of a population with N individuals is no longer described by a deterministic function n(t), but by a probability pu(t) = Pr{N(t) = n}. Note in the above formula, the N is an integer-valued random variable, which can take any values from 0,1,2,...,n,..., with the probability Pn(t). In other words, we no longer talk about precisely the number of individuals at time t, instead we talk about the probability of having n number of individuals in the population at time t. If all the individuals in the population are statistically independent and identical distributed (i.i.d.), with constant per capita birth rate b and death rate d, then Pn(t) satisfies the Equation (8.2) (a) Explain why is the coefficient of the Pn-1 term is (n-1), and coefficient of the Pn+1 term is (n+1)d. Also explain what the two positive terms, b(n-1)Pn-1 and d(n+1)Pn+1 represent, and what about the two negative terms bnp and dnp,? (b) The Equation (8.2) is really a system of ODEs for po(t), p.(t), p2(1), ...,Pn(t),..., Write out the first two equations of the system: dpo() dt and dpi(t) dt explain each and every term on the right-hand sides. ..

(c) Assuming that the order of summation and derivative can be exchanged, i.e., d (dpa (t) dt dt 1 (Σο) - Σ() (( (() + P (1) +...) = 0. show that Explain the meaning of this mathematical equation.

8.1 Simple population kinetics with birth and death Let us consider a single population, with a constant per capita birth rate u and death rate v. Therefore, the conventional differental equations for the population size z at time t is 1 dx(t) =b-d. (8.1) dt In the classic population biology, this equation is true simply because the definition of per capita birth and death rates! But we now know that this equation can not be really true! The popuation size does not grow by a continuous value, it has to be by integers; and furthermore, the per capita birth and death rates are only “average” concepts. There are a great deal of randomness in the birth and death processes. Therefore, instead of using real valued a(t) as the population size at time t, we shall now study the probability of the population having n individuals at time t: Pn(t). Then this Pn(t) satisfies the following equation: dpr(t) (8.2) dt = (n - 1)bpn-1(t) - (b+d)npn(t) + (n + 1)dpn+1(t). The assumption here is that all the individuals in the population, though are different, they are statistically identical, and independent in their giving birth and going death; and the birth and death events for a person has an exponential distributed time. The population size n as a function of t is random, or stochastic, as shown in Fig. 8.1. The Eq. 8.2 looks complicated; but we can have the following two important results. First, Pn(t) {[(n − 1)bpn-1(t) – (6 + d)npu(t) + (n + 1)dpn+1 (t)] dt 0. (8.3) This is of course expected: The total probability should always be 1 and this does not change with time. n=0 n=1