Recursive Estimation of a Discrete-Time Lotka-Volterra Predator-Prey Model

ةصلاخ : نـم نيعوـنل يناويحلا دادعتلا باسح ةلأسم يهو ينيبلا ءاصحلإا يف ةمهم ةلأسم جلاعي لاقملا اذه ةداطصملا و ةدئاصلا يهو تاناويحلا . ي مث ةصاخ ةراشإب زيمتو ةيئاوشع ةنيع ذخؤت ةيعون لك نم اهحارس قلط عيطقلا ةيقب لخاد . ةصاخلا ةراشلإاب نيزيمملا دارفلأا اهنم يصحيو نيعيطقلا نم ةيئاوشع ةنيع ذخؤت ةرتف دعبو ةدئاـصلا تاـناويحلا ةـعومجمل ررـكتملا يطرشلا عيزوتلا ريدقتل سايقلا قرط لمعتست تامولعملا هذه نمو ةداطصملاو .   ABSTRACT: In this paper, using hidden Markov models, we estimate the number of individuals in a two-species (predator-prey) animal population using partial information provided by the so-called capture-recapture technique. Random samples of individuals are captured, tagged in some way and released. After some time other random samples are taken and the marked individuals are observed. Using this information, we estimate (recursively) the sizes of the two populations. Also, using the Expectation Maximization (EM) algorithm, the parameters of the model are updated.


Introduction
ne of the first models to incorporate interactions between predators and preys was proposed in 1925 by the American biophysicist Alfred Lotka and the Italian mathematician Vito Volterra .Vito Volterra (1860Volterra ( -1940) ) was a famous Italian mathematician who retired from a distinguished career in pure mathematics in the early 1920s.His son-in-law, Humberto D'Ancona, was a biologist who studied the populations of various species of fish in the Adriatic.Volterra developed a series of models for interactions of two or more species.

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Alfred J. Lotka (1880Lotka ( -1949) ) was an American mathematical biologist (and later actuary) who formulated many of the same models as Volterra, independently and at about the same time.His primary example of a predator-prey system comprised a plant population and an herbivorous animal dependent on that plant for food.
Here we consider a discrete-time stochastic version of the Lotka-Volterra model.Hidden Markov models (Elliott et al. 1995) have been used extensively in many areas of science and technology.In this paper we are extending the use of this powerful tools (see Aggoun and Elliott 1998 for a single species model) to estimate the hidden number of individuals in a multi-species animal population using partial information provided by the so-called capture-recapture technique (see Seber 1982, for instance).
Two random samples of individuals one from the predator population and one from the prey population are captured, tagged or marked in some way, and then released.After allowing time for the marked and unmarked to mix sufficiently, two second simple random samples from both populations are taken and the marked ones are observed.At epoch write Z for the prey population size, z for the number of marked and released preys, 1 k k z z = = ∑ for the total number of captured and marked preys up to time k , R for the sample size, z for the number of available marked individuals for sampling and Z y for the number of captured (or recaptured) marked individuals.
Similarly write X for the predator population size, x for the number of marked and released predators, for the total number of captured and marked predators up to time k , F for the sample size, x for the number of available marked predators for sampling and X y for the number of captured (or recaptured) marked predators.

Model Assumptions and Recursive Estimation
All random variables are defined initially on a probability space ( ) F P Ω, , .All the filtrations defined here are assumed to be complete.

Write (
) , , , , , , , ≤ , and ( , ) k X and k Z represent the number of predators and preys, respectively, that are alive at time period k, then a (discrete time) Lotka-Volterra type model is: where the parameters are defined by: a is the efficiency of turning predated preys into predators. Proof.
However using repeated conditioning it is enough to show that . The point here is that: Proof.We shall check the claim for the three processes k Z , k z and Z k y .For any integrable real-valued functions f , g and h and using a version of Bayes' theorem (see Elliott et al. 1995) we can write: That is, under Q the three processes are independent sequences of random variables with the desired distributions.
Using this fact we derive a recursive equation for the unnormalized conditional distribution of k Z and k X For any measurable test function f consider: The denominator of (8) being a normalizing factor we focus only on the expectation under Q in the numerator.Write Theorem 1.The unnormalized conditional joint probability density function of the populations' sizes given by the dynamics in (1) and (2) follows the recursion: where (Note we take 0 0 1 = .) Proof.In view of Lemma 2 the left hand side of ( 9) is: Comparing this last expression with the right hand side of (9) gives the result.
If the distribution of , is a delta function concentrated at ( ) A B , say

Parameter Revision
Here we shall assume, for simplicity, that the random variables v and w in our model are standard normal (means 0 and standard deviations 1).The EM algorithm, (Baum andPetrie 1966, Dempster et al. 1977) is a widely used iterative numerical method for computing maximum likelihood parameter estimates of partially observed models such as linear Gaussian state space models.For such models, direct computation of the MLE is difficult.The EM algorithm has the appealing property that successive iterations yield parameter estimates with nondecreasing values of the likelihood function.
Suppose that we have observations 1 K y … y , , available, where K is a fixed positive integer.Let { } , ∈Θ be a family of probability measures on ( ) F Ω, , all absolutely continuous with respect to a fixed probability measure 0

P . The log-likelihood function for computing an estimate of the parameter θ based on the information available in
, and the maximum likelihood estimate (MLE) is defined by ˆargmax ( ) Let 0 θ be the initial parameter estimate.The EM algorithm generates a sequence of parameter estimates { } j θ , 1 j ≥ , as follows.Each iteration of the algorithm consists of two steps: Step 1. (E-step).Set ˆj θ θ = and compute ( ) Step 2. (M-step).Find (As a consequence of ( 1), ( 2) and (iii), clearly ( ) , is continuous in both θ and ˆj θ .
Then by Theorem 2 in Wu (1983), the limit of the sequence EM estimates { } j θ has a stationary point θ for some stationary point θ .To make sure that t L is a maximum value of the likelihood, it is necessary to try different initial values 0 θ .
The model in ( 1) and ( 2) is determined by the parameters a , b , c and d which need to be updated as new information is obtained.These parameters are estimated using the expectation maximization (EM) algorithm.
Maximum likelihood estimation of the parameters via the EM algorithm requires computation of the filtered estimates of quantities such as ∑ First we compute ML estimates of the parameters , is derived.
To update the set of parameters from θ to θ , the following density is introduced where ( ) R θ does not involve θ .
To implement the M-step set the derivatives 0

∑
For any "test" function g , write The first expectation is simply 12) and ( 13) yields the result. Write To update the parameter from X p to X p , the following density is introduced

∫ ∫
Proof.First note that The first expectation is simply 1 are given by the following recursions.
b is the natural death rate of predators in the absence of food (preys), c is the natural growth rate of preys in the absence of predation, d is the death rate per encounter of preys due to predation, v and w are sequences of independent random variables with some (either discrete or continuous with finite supports) densities k φ and k ϑ respectively and 1 σ , 2 σ are some positive real numbers.The random variables v and w indicate other sources of variations in the populations like death caused by old age,