Itô versus Stratonovich calculus in random population growth
Introduction
Let N = N(t) be the size (number of individuals, biomass, density) at time t ⩾ 0 of a population of animals, bacteria or cells (in an organism or an organ). We assume that the population lives in an environment subjected to random fluctuations. We shall refer to by total growth rate and to the per capita growth rate simply by growth rate. We can model the dynamics by assuming that the growth rate is the sum of an ‘average’ growth rate g(N) (a deterministic and usually density-dependent component) and perturbations caused by the random environmental fluctuations. Assuming a small correlation time for such perturbations, we can approximate them by a white noise σε(t), where σ > 0 is the (per capita) noise intensity and ε(t) is a standard continuous-time white noise (a generalized stationary non-correlated Gaussian process, formally the generalized function derivative of the non-differentiable standard Wiener process w(t)). As usual, we have abbreviated N(t) = N(t, ω) and w(t) = w(t, ω), not writing explicitly the dependence of these stochastic processes on chance ω ∈ Ω, where Ω is the set of all possible random environmental scenarios endowed with a probability space structure . By a scenario ω we mean a specific possible arrangement of what environmental conditions a population might experience from time t = 0 to time t = ∞. Each fixed scenario ω defines a trajectory (function only of time). Usually, we only observe one trajectory, the one corresponding to the environmental scenario ‘assigned by chance’ to the particular population we are observing.
We obtain the stochastic differential equation (SDE) modeland assume the initial population size N0 to be known. If σ was = 0 instead of being positive, we would have a deterministic ordinary differential equation (ODE) model.
We can also consider a generalization, where we allow the noise intensity σ to be density-dependent (σ(N)) instead of being constant.
Models of this sort have been proposed in the literature, and their properties studied for specific growth rate functions [like the logistic g(N) = r(1 − N/K), the Gompertz g(N) = r ln(K/N), and several others] and specific noise intensity functions [usually σ(N) ≡ σ, but sometimes also σ(N) = cg(N)]. [1] was the pioneer work but many others have dealt with the subject in the following years like, for instance, [2], [3], [4], [5], [6], [7] and references therein. But one could not be sure whether the properties obtained for these specific growth rate function models were valid only for the specific cases or were indeed general properties of populations growing in a random environment. This is a real problem since, with the available evidence, the correctness of any specific growth function is far from being established. So, it would be nice to obtain similar results for a general model with an arbitrary growth function satisfying only mild regularity assumptions and some general assumptions dictated by biological considerations. This was done in [8]. In [9] these results were further generalized to arbitrary noise intensities functions σ(N) satisfying some reasonable assumptions. Probabilities of passing by low or by high thresholds and their first passage times are relevant for extinction studies, environmental protection and pest control; these issues can be seen, for a certain class of models, in [10] and references therein.
Similar models with harvesting have been studied for specific growth rate, noise intensity, and harvesting rate (h(N)) functions. See, for example, [11], [12], [13], [14], [15], [16]. In [17] the main results were generalized to arbitrary growth rate and harvesting rate functions, assuming constant noise intensity. In [18] they were further generalized to arbitrary noise intensities. For the question of optimal harvesting studied as a stochastic control problem, one can see for example [19], [20], [21].
The issue of parameter estimation, testing, and prediction based on discrete observations on the single trajectory usually available can be seen in [6], [14], [22], [23].
Here, we will resolve, with an interesting interpretation, the long standing controversy of which stochastic calculus, Itô or Stratonovich, is more appropriate in dealing with the general model (1) when modeling population growth in a random environment. The further generalization to an even more general SDE model where the noise intensity, instead of being constant, is allowed to be a density-dependent function σ(N), will be treated in a forthcoming paper. Let me anticipate, though, that the conclusions for this density-dependent noise intensity case are very similar to the ones obtained here, requiring only slight adaptations in the interpretation, although the mathematical derivations become quite more technical.
The controversy comes from the fact that the two stochastic calculi do not yield the same solution to the SDEs. Let us look at the origin of this problem. The SDE model (1) can be written in the formwhereis the ‘average’ total growth rate when population size is N andis the total noise intensity (intensity of the effect of random environmental fluctuations on the total growth rate dN/dt) when population size is N. The SDE can also be written as an equivalent stochastic integral equationAssuming adequate regularity conditions, the first integral can, for each trajectory (i.e., for fixed ω), be defined as a Riemann integral. However, the second integral cannot be defined as a Riemann–Stieltjes integral since the Wiener process w(t) is a function almost surely of unbounded variation. There are many possible definitions of this stochastic integral, but the ones commonly used in the literature are the Itô and the Stratonovich integrals. The Itô integral has nice probabilistic properties (it has zero expectation and a convenient expression for its variance, besides being a martingale as a function of t) but does not satisfy ordinary rules of calculus. A new stochastic calculus (Itô calculus) had to be developed, in which the chain rule of differentiation is different from the usual one. The Stratonovich integral does not have so nice probabilistic properties but the corresponding calculus, the Stratonovich calculus, satisfies the usual rules of calculus (including the usual chain rule of differentiation). In Section 2, we briefly describe the differences between the two calculi.
According to which calculus one uses (i.e., according to the type of integral one uses in interpreting the equivalent stochastic integral equation (2)), we have different solutions to the SDE (1). One wonders which results are the correct ones and there is controversy in the literature over which calculus is indeed appropriate to model population growth in a random environment. In Section 3 we refer to that controversy and illustrate the consequences for the population behavior and for some ecological theories of the differences between the two calculi. For instance, even in the simple example of density-independent growth, where we have a constant ‘average’ growth rate (g(N) ≡ r), the consequences are not only quantitative. One important qualitative difference is that, using Stratonovich calculus, extinction would (similarly to the deterministic model) occur with probability one if the ‘average’ growth rate r is <0, but, with Itô calculus, one can have extinction with probability one even for positive values of the ‘average’ growth rate r (if r < σ2/2). We also speak about the main recommendations one can find in the literature on which circumstances to use Itô calculus and on which circumstances to use Stratonovich calculus. Such recommendations are, however, not helpful in the majority of the cases.
That lead us to try to shad some light on the issue in [24]. However, the parametric approach taken there was quite limited. Although appropriate to resolve the problem in the density-independent growth model and although it could be used asymptotically as t → ∞ for a certain class of density-dependent models, it was not appropriate for most density-dependent models. Here, we take an approach that, although founded on the same basic idea, is completely different. This new approach works for all density-dependent models and completely and exactly (not just asymptotically) elucidates the difference between the two calculi and interprets it in terms of the dynamics of the population. It also exactly solves in all cases the problem of which calculus to use and how to use it.
In Section 4, we illustrate the basic idea behind our new method using the simple example of density-independent growth. We show that, while one may be using the same letter ‘r’ in both calculi to refer to the ‘average’ growth rate (as it is common in the literature), the interpretation of ‘r’ in terms of population dynamics is completely different under the two calculi. Indeed, we will show that, under Itô calculus, r indeed means the arithmetic average growth rate and, under Stratonovich calculus, r indeed means the geometric average growth rate (which is the right average to use for comparison purposes with the deterministic case, since population growth has a multiplicative type of dynamics). Taking into account the difference between the two averages, the two calculi give exactly the same results after all. In particular, both calculi give extinction with probability one when the geometric average growth rate is negative. The apparent difference between the two calculi was merely semantic and due solely to the wrong implicit assumption that the letter ‘r’ meant the same thing under the two calculi, the so-called ‘average’ growth rate. That wrong implicit assumption prevails in the literature, which also implicitly assumes that the term ‘average’ (the quotes are ours), without specification of what type of average, is unequivocal and therefore that the same ‘average’ is used in both calculi. This is not true and it is the source of the whole controversy.
In Section 5, we extend these results for the general model (1), again showing that the ‘average’ growth rate g(N) indeed means an arithmetic or a geometric average according to whether one uses Itô or Stratonovich calculus. Again, taking into account the difference between the two averages, the two calculi give exactly the same results. In particular, under certain biologically reasonable conditions on g(N), extinction will occur with probability one for negative geometric average growth rates at low population densities (more exactly, at the zero limit population density). The forthcoming paper that further generalizes to the case of density-dependent noise intensities σ(N) reaches similar conclusions, but the interpretation of g(N) under the Stratonovich calculus is slightly different: it is a modified geometric average growth rate.
Section 6 presents the main conclusions, particularly exact recommendations on which calculus to use and how to use it. It turns out that it does not matter which calculus one uses, as long as one uses for the expression of the ‘average’ growth rate g(N) the expression that correctly describes the appropriate average for that calculus. Namely, for the general model (1), one should use the arithmetic average if one uses Itô calculus and the geometric average if one uses Stratonovich calculus. If one adopts such recommendation, the two calculi give exactly the same results.
Section snippets
Itô and Stratonovich calculi
As already mentioned, the SDE (1) is equivalent to the stochastic integral equation (2), but the second integral in (2) cannot be defined, for a fixed trajectory, as the classical Riemann–Stieltjes integral. In fact, for sequences of decompositions 0 = t0,n ⩽ t1,n ⩽ ⋯ ⩽ tn,n (n = 1, 2, …) with diameters converging to zero, the Riemann–Stieltjes sumshave different mean square (m.s.) limits depending on the choice of the intermediate points τi,n ∈ [ti−1,n, ti,n]. Among the many
The controversy
To illustrate the consequences of the difference between the two calculi, let us consider the malthusian (density-independent) growth in a random environment. It corresponds to a constant ‘average’ growth rate g(N) ≡ r. To make the distinction between the two calculi more transparent we will use ri for the Itô calculus and rs for the Stratonovich calculus. There is no need to distinguish between the noise intensities because they play an identical role in both calculus.
Let us start by studying
The resolution for density-independent growth
Let us go back to the density-independent growth model (10) or (13), respectively for Stratonovich and for Itô calculus, considered for illustration purposes in the previous section. The model corresponds to a constant g(N). Although, in the literature on the Itô-Stratonovich controversy, the same letter ‘r’ is used in both models to represent an implicitly common ‘average’ growth rate, we did use ‘rs’ and ‘ri’ to make the distinction between the two calculi more transparent. We also asked the
The resolution for density-dependent growth
We will now consider the more general model (1) with density-dependent ‘average’ growth rate g(N). To make more transparent the distinction between the Itô and the Stratonocich calculus we will use gi(N) and gs(N), respectively. There is no need to distinguish between the noise intensities because they play an identical role in both calculus.
Let us first consider the Itô calculus and write the SDE in the standard form:We will assume that
- (A)
gi(·) : (0, +∞) ↦
Conclusions
In the context of density-dependent (and also density-independent) SDE population growth models in a randomly fluctuating environment (as well as in other contexts), there was a long standing controversy of whether it is more appropriate to use Itô or Stratonovich calculus. The two calculi lead to qualitatively different results on extinction and existence of a stationary density (a kind of stochastic equilibrium). For instance, for g strictly decreasing, while under Stratonovich calculus
Acknowledgment
This work was performed at CIMA-UE (Centro de Investigação em Matemática e Aplicações da Universidade de Évora), a research center financed by the Portuguese research funding agency FCT (Fundação para a Ciência e a Tecnologia) within its ‘Programa de Financiamento Plurianual’ (under FEDER funding).
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