Refereed scientific paper on stochastic differential equation models of population growth in random environments with resolution of the controversy on the use of Itô or Stratonovich calculus (extension to density-dependent noise intensities). The paper is in press in the Bulletin of ISI containing the Proceedings of the 56th Session of the ISI (2007). An electronic version is available

Braumann, Carlos A.

Repositório Científico da Universidade de Évora

    Population Growth in Random Environments: Which Stochastic Calculus?  Carlos A. Braumann      Bulletin of the International Statistical Institute, LXII (Proc. 56th Session of the International Statistical Institute) In press Population Growth in a Random Environment: Which StochasticCalculus?Braumann, Carlos A.Universidade de E´vora, CIMA (Centro de Investigac¸a˜o em Matema´tica e Aplicac¸o˜es)Rua Roma˜o Ramalho 597000-671 E´vora, PortugalE-mail: braumann@uevora.pt1. IntroductionLet N = N(t) be the size (number of individuals, density, biomass) of a population (of animals,plants, bacteria, cells) at time t ≥ 0 and assume the initial population size N(0) = N0 > 0 is known. Ageneral density dependent growth model assumes that the per capita growth rate (abbrev. growth rate)has the form 1NdNdt = f(N), where f(·) : (0,+∞) 7→ (−∞,+∞) is a continuously differentiable functionsuch that the limit f(0+) := limN↓0 f(N) exists (may be infinite) and F (0+) := limN↓0 F (N) = 0 (nospontaneous generation), where F (N) = Nf(N) is the total growth rate of the population.In a randomly varying environment, the growth rate 1NdNdt will have an ”average” value f(N) andrandom perturbations that we approximate by σε(t), where ε(t) is a standard white noise (formallythe generalized function derivative of the standard Wiener process W (t)) and σ is the noise intensity.The resulting stochastic differential equation (SDE) model is 1NdNdt = f(N) + σε(t) ordN(t) = F (N(t))dt + V (N(t))dW (t),(1)where V (N) = σN . Equation (1) is equivalent to the stochastic integral equationN(t) = N0 +∫ t0F (N(s))ds +∫ t0V (N(s))dW (s).(2)We will assume also that f , besides the above assumptions, is such that the boundaries N = 0 andN = +∞ are unattainable (so that the solution of the SDE exists, is unique and has values in (0,+∞))and the moments required in this paper do exist (see Braumann, 2007a, for mild sufficient conditions).Models of this sort have been proposed in the literature with specific forms of the function f .Levins (1969) was the pioneer work. Beddington and May (1977) launched the harvesting models,where a harvesting term is added to the growth equations. A list of references can be seen in Braumann(1999a,b), where, for non-harvesting and harvesting models respectively, the properties of the generalmodel with arbitrary f (satisfying only reasonable assumptions) were studied, including conditionsfor non-extinction and for the existence of a stationary probability density.There is, however, a problem with these models. The second integral in (2) cannot be defined asa classical Riemann-Stieltjes integral because the Wiener process W (t) is of unbounded variation. Ifwe consider a sequence of decompositions 0 = t0,n ≤ t1,n ≤ ... ≤ tn,n = t (n = 1, 2, ...) with diametersconverging to zero, the Riemann-Stieltjes sumsn∑i=1V (N(τi,n)) (W (ti,n)−W (ti−1,n)) have differentmean square (m.s.) limits depending on the choice of the intermediate points τi,n ∈ [ti−1,n, ti,n].Among the many possible choices, two stand out in the literature.One is the non-anticipative choice τi,n = ti−1,n that defines the Itoˆ integral. Itoˆ calculus has niceprobabilistic properties but does not satisfy ordinary rules. In particular, it satisfies a different chainrule of differentiation. Namely, if Y (t) = h(t,N(t)), with h(t, x) of class C1,2, we get(I) dY =(∂h(t,N)∂t+∂h(t,N)∂xF (N) +12∂2h(t,N)∂x2V 2(N))dt+∂h(t,N)∂xV (N)dW(3)instead of the usual rule (applicable to Stratonovich calculus)(S) dY =(∂h(t,N)∂t+∂h(t,N)∂xF (N))dt+∂h(t,N)∂xV (N)dW.(4)We have used ”(I)” or ”(S)” to distinguish between the Itoˆ and the Stratonovich calculi. The Strato-novich integral is, under adequate regularity conditions, the m.s. limit ofn∑i=1(V (N(ti−1,n)) + V (N(ti,n))2)(W (ti,n)−W (ti−1,n)) .This integral anticipates (”guesses”) a bit into the future and does not have the nice probabilisticproperties of the Itoˆ integral. The Itoˆ and Stratonovich calculi are the ones commonly used in theliterature. For more details on them, see, for instance, Arnold (1974) or Øksendal (2003).The problem is that the solutions of SDE depend on the stochastic calculus used. For instance,when f(N) ≡ r (Malthusian model), we have, under Stratonovich calculus, that extinction occurswith probability one if the ”average” growth rate r is negative and extinction has zero probability ofoccurring if r is positive. This behavior is similar to the deterministic case (σ = 0). However, if oneuses Itoˆ calculus, extinction occurs with probability one when r < σ2/2. Will a population with small(smaller than σ2/2) positive ”average” growth rate r be extinct ir not?The answer depends on the calculus used and this is a source of controversy and mistrust.The same question can be asked for a general strictly decreasing function f , since Braumann(1999a) has shown, for Stratonovich calculus, that extinction occurs with probability one if f(0+)(”average” growth rate at low population sizes) is negative and occurs with zero probability (thereis even a stationary probability density) if f(0+) is positive. However, for Stratonovich calculus thecriteria is whether f(0+) is smaller or larger than σ2/2.There are recommendations, based on some limit theorems, on which calculus to use dependingon whether generations are discrete and noise is white in discrete time (Itoˆ calculus) or generationsare continuous and noise is slightly colored (Stratonovich calculus) but reality is more complex thanthat. A paper resolving partially the controversy in the asymptotic regime is Braumann (1983). Theresolution of the controversy in the Malthusian example just mentioned is in Braumann (2003). Thefull resolution of the controversy for models of type (1) with arbitrary f is in Braumann (2007a)(see also Braumann, 2007b, for the case of harvesting models), as well as references on the history ofthe controversy. A brief account is made on Section 2. However, we have only considered the caseof constant noise intensity σ. Here, in Section 3, we extend the resolution to cases where the noiseintensity σ(N) might vary with population size.2. The resolution of the controversy for constant noise intensityWe will show that the controversy is due to the wrong presumption that f(x), taken as the”average” growth rate when population size has size x, means the same average under the two calculi.To avoid such semantic confusion upon which rests all the controversy in the literature, we will use fiand fs to denote f according to whether we use Itoˆ or Stratonovich calculus.Of course, for the deterministic model dNdt = F (N) = Nf(N), the (per capita) growth rate R(x)when population size is x at time t, is by definition R(x) := 1x(dNdt)N=x= 1x lim∆t↓0N(t+∆t)−x∆t = f(x).However, for the stochastic models, N(t+∆t) is a random variable and so is R(x). So, we lookfor an average growth rate. Let us consider the arithmetic average, which is the usual expected value.Of course, since we are considering the situation that at time t the population size is x, we shouldtake the expectation conditioned on that knowledge. Let us denote it by Et,x[...] = E[...|N(t) = x].Then, the arithmetic average growth rate when population size is x at time t, is defined byRa(x) :=1xlim∆t↓0Et,x[N(t+∆t)]− x∆t.(5)We could, however, consider the geometric average growth rate defined byRg(x) :=1xlim∆t↓0exp (Et,x[ lnN(t+∆t)])− x∆t.(6)Consider the Itoˆ and the Stratonovich SDE(I) dN(t) = fi(N(t))N(t)dt + σN(t)dW (t)(7)(S) dN(t) = fs(N(t))N(t)dt + σN(t)dW (t).(8)The solutions (see Arnold, 1974) are homogeneous diffusion processes with common diffusion coefficientb(x) := lim∆t↓0Et,x[(N(t+∆t)− x)2]∆t= V 2(x) = σ2x2(9)and drift coefficients, respectivelyai(x) := lim∆t↓0Et,x[N(t+∆t)− x]∆t= fi(x)x(10)as(x) := lim∆t↓0Et,x[N(t+∆t)− x]∆t= fs(x)x+14db(x)/dx = (fs(x) + σ2/2)x.(11)Therefore, from (5), (10), and (11), we obtain the arithmetic average growth rate when popula-tion size is x at time t, respectively for the Itoˆ SDE (7) and the Stratonovich SDE (8):Ra(x) =1xai(x) = fi(x)(12)Ra(x) =1xas(x) = fs(x) + σ2/2.(13)If one makes the change of variable Y = lnN , applying to equations (7) and (8) the chain rules(3) and (4), one obtains (I) dY = (fi(eY ) − σ2/2)dt + σdW (t) and (S) dY = fs(eY )dt + σdW (t),respectively. So, with y = lnx, the drift coefficients lim∆t↓0Et,x[Y (t+∆t)−y]∆t for Y are, respectively,fi(ey) − σ2/2 and fs(ey). Therefore, from (5), one obtains the geometric average growth rate whenpopulation size is x at time t, respectively for the Itoˆ SDE (7) and the Stratonovich SDE (8):Rg(x) = fi(x)− σ2/2(14)Rg(x) = fs(x).(15)Conclusion: Contrary to what has been presumed in the literature, f(x) means two different”average” growth rates under the two calculi. It is the arithmetic average growth rateunder Itoˆ calculus and the geometric average growth rate under Stratonovich calculus.Taking into account the difference between the two averages, the results of the two calculi completelycoincide. In fact, the apparently different solutions of the Itoˆ SDE (7) and the Stratonovich SDE (8)are indeed the same, namely the homogeneous diffusion process with diffusion coefficient (9) and driftcoefficient xRa(x). They looked different because, instead of using a concrete average growth rate, wewere expressing them in terms of an unspecified ”average” wrongly assumed to be the same averageunder the two calculi. For the particular case of strictly decreasing growth rate f , extinction will occurfor both calculi when the geometric average growth rate at low population sizes Rg(0+) is negative.3. The case of density-dependent noise intensitiesWe now consider the generalization to a density-dependent noise intensity σ(N), assumed to bea positive continuously differentiable function for N > 0 such that σ(0+) exists and V (0+) = 0, whereV (N) = σ(N)N . The diffusion and drift coefficients are nowb(x) = V 2(x) = σ2(x)x2(16)ai(x) = fi(x)x(17)as(x) = fs(x)x+14db(x)dx=(fs(x) + σ2(x)/2 + xσ(x)σ′(x)/2)x.(18)We obtain for the arithmetic average growth rate when population size is x at time tRa(x) = ai(x)/x = fi(x)(19)Ra(x) = as(x)/x = fs(x) + σ2(x)/2 + xσ(x)σ′(x)/2,(20)respectively for the Itoˆ SDE and the Stratonovich SDE. So, the arithmetic average growth rate is stillfi(x) for the Itoˆ SDE. However, fs(x) is no longer the geometric average growth rate.The function φ(x) =∫ xc1zσ(z)dz (where c is an fixed arbitrary positive constant) is strictlyincreasing for x positive, and so it has an inverse φ−1. Let us consider the φ-average growth rateRφ(x) :=1xlim∆t↓0φ−1 (Et,x[φ(N(t+∆t))])− x∆t.(21)Notice that, when σ(x) is a constant σ, this is just the geometric average.Under Stratonovich calculus, Y = φ(N) satisfies the SDE (S) dY = fs(φ−1(Y ))σ(φ−1(Y )) dt + dW (t), andso, in terms of Y , the drift coefficient is lim∆t↓0Et,x[Y (t+∆t)−y]∆t =fs(φ−1(y))σ(φ−1(y)) , where y = φ(x). Therefore,Et,x[Y (t + ∆t)] = y +fs(φ−1(y))σ(φ−1(y)) ∆t + o(∆t). Apply φ to both sides, expand about y and notice thatdφ−1(y)dy =1dφ(x)/dx = xσ(x) to obtain φ−1 (Et,x[Y (t+∆t)]) = x+ xfs(x) + o(∆t). From (21) we getRφ(x) = fs(x).(22)Thus, for Stratonovich calculus, fs(x) is the φ-average growth rate. Again, taking into accountthe difference between the arithmetic and the φ-average, the results of the two calculi coincide.4. ConclusionWe have extended the resolution of the controversy on whether to use Itoˆ or Stratonovich calcu-lus when modeling population growth in a random environment to the case of density-dependent noiseintensities. Again we show that what was thought to mean the ”average” growth rate meant really dif-ferent types of averages, the arithmetic average under Itoˆ calculus and a φ-average under Stratonovichcalculus (coinciding with the geometric average for the case of constant noise intensity). Taking intoaccount the difference between the two averages, the two calculi give completely coincidental results.Acknowledgment. C. A. Braumann is a member of CIMA, a research center financed by FCT (Fundac¸a˜opara a Cieˆncia e a Tecnologia) within its ’Programa de Financiamento Plutianual’ (under FEDER funding).REFERENCESArnold, L. (1974). Stochastic Differential Equations: Theory and Applications. Wiley, New York.Beddington, J. R. and May, R. M. (1977). Harvesting natural populations in a randomly fluctuatingenvironment. Science 197: 463-465.Braumann, C. A. (1983). Population growth in random environments. Bull. Mathem. Biol. 45: 635-641.Braumann, C. A. (1999a). Applications of stochastic differential equations to population growth. In:Bainov, D. (Ed.), Proc. Ninth International Colloquium on Differential Equations, VSP, Utrecht, pp. 47-52.Braumann, C.A. (1999b).Variable effort fishing models in random environments. Math.Biosci. 156: 1-19.Braumann, C. A. (2003). Modeling population growth in random environments: Ito or Stratonovichcalculus? Bull. International Statistical Institute LX (CP1): 119-120.Braumann, C. A. (2007a). Itoˆ versus Stratonovich calculus in random population growth. Math. Biosci.206: 81-107.Braumann, C. A. (2007b). Harvesting in a random environment: Itoˆ or Stratonovich calculus? J.Theoretical Biology 244: 424-432.Levins, T. (1969). The effect of random variations of different types on population growth. Proc. Natl.Acad. Sci. USA 62: 1062-1065.Øksendal, B. (2003). Stochastic Differential Equations: An Introduction with Applications (sixth edition).Springer, Berlin.

Population growth in random environments: which stochastic calculus?

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Population growth in random environments: which stochastic calculus?

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