In these notes we review some recent results on the homoenergetic solutions for the Boltzmann equation obtained in [
Citation: Alessia Nota, Juan J. L. Velázquez. Homoenergetic solutions of the Boltzmann equation: the case of simple-shear deformations[J]. Mathematics in Engineering, 2023, 5(1): 1-25. doi: 10.3934/mine.2023019
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In these notes we review some recent results on the homoenergetic solutions for the Boltzmann equation obtained in [
Relatively large areas in the western region of the United States are classified as arid or semi-arid environments, which are characterized in part by their limited and variable precipitation. Semi-arid regions are expected to receive around 10 to 30 inches of average annual precipitation (
Extensive mathematical modeling and analysis of semi-arid water-vegetation systems emerged for at least the past twenty years, especially since the appearance of deterministic ecohydrological models with broad developments focused on vegetation pattern formation, see for instance [5,12,19,22,25,26,27]. The effects of noise on dryland ecosystems that are usually described by deterministic models showing bistability have also been analysed, [6,21], suggesting the possibility of creation or disappearance of vegetated states in the form of noise-induced transitions, [11]. In contrast, the goals in this paper are to present a stochastic differential equation approximation for an idealized water-vegetation (non-spatial) discrete system and the estimation of mean transition times into a desert state. We emphasize that our model is not obtained by adding "noise" to a differential equation as previously done [6,21], but by the construction of a diffusion approximation.
As a first step, we set up a Markov jump process that incorporates the interactions in an idealized water-vegetation system. Similar conceptual models have been used successfully in other biological contexts, see for instance [17] and [18,15]. The model involves only water and vegetation biomass, in an environment of limited capacity. When this capacity (or "system size") increases it gives rise to a deterministic system of differential equations for the mean densities. We deduce an intermediate mesoscale stochastic model between the jump process and the differential equations obtained for the means. Using estimated parameters for vegetation and precipitation in semi-arid landscapes from the literature, and data for state precipitation anomalies in California as baseline, we estimate the mean times for a system to reach desertification in a range of realistic precipitation anomalies, i.e. departures from long term mean. With these results we finally quantify, for this simple model, the dependence between changes in precipitation anomalies and mean transition times to the desert state.
We start by defining a Markov jump process that represents a simplified version of the real interactions between water and vegetation at a small scale. We do this through the discretization of (alive) vegetation biomass and water volume in small units (individuals), for which a specific set of stochastic events can be explicitly characterized. Naturally, as we transition into larger scales discreteness is lost, and the continuous state space takes place.
We first consider a patch with finite capacity, say
Event | Transition | Jump | Jump rate |
Vegetation biomass loss | |||
Incoming water | |||
Water evaporation | |||
Increase vegetation by | |||
water take up | |||
It is then straightforward to find the probability rates of transition from a state
(a)
(b)
(c)
(d)
Using these rates we can write the associated Kolmogorov equation (see [7] for instance),
dP(n,m,t)dt=T(n,m|n−1,m+1)P(n−1,m+1,t)+T(n,m|n+1,m)P(n+1,m,t)+T(n,m|n,m+1)P(n,m+1,t)+T(n,m|n,m−1)P(n,m−1,t)−(T(n+1,m−1|n,m)+T(n−1,m|n,m)+T(n,m−1|n,m)+T(n,m+1|n,m))P(n,m,t). |
where
d⟨n⟩dt=N∑n,m=0[T(n+1,m−1|n,m)−T(n−1,m|n,m)]P(n,m,t)=b⟨n⟩N⟨m⟩N−1−d⟨n⟩N, | (1) |
where the correlations between the random variables are neglected under the assumption of a large
dρvdt=˜bρvρw−˜dρv, | (2) |
where
d⟨m⟩dt=N∑n,m=0[T(n,m+1|n,m)−T(n,m−1|n,m)]P(n,m,t) =s−s⟨n+m⟩N−b⟨n⟩N⟨m⟩N−1−v⟨m⟩N | (3) |
Similarly, writing the mean density of water by
dρwdt=˜s(1−(ρv+ρw))−˜bρvρw−˜vρw, | (4) |
where
dρvdt=˜bρ2vρw−˜dρv | (5) |
instead of equation (2). A similar change happens in equation (4), which is now
dρwdt=˜s(1−(ρv+ρw))−˜bρ2vρw−˜vρw. | (6) |
Equations (5) and (6) constitute a system of differential equations that serves as approximation to the dynamics of the mean densities for large values of
The diffusion approximation to our model (the mesoscale model) describes the system as an intermediate approximation that emerges between the Markov jump process model (microscale model) and the differential equations for the mean densities (macroscale model). For this approximation the state variables are continuous but include random fluctuations. We expect the new model to incorporate the differential equations and terms that express random fluctuations around the mean densities.
To obtain a representation of the Markov jump process as a diffusion process one can follow either Kurtz's method [14], or find the same equations via the Fokker-Planck equation [18,7]; see [4] for a nice introduction. In Kurtz's approach, which is the one we use here, the jump process is represented by
X(t)=X(0)+∑rrN(r)(∫t0NΦ(1NX(s);r)ds), | (7) |
where
dY(t)=∑rrΦ(Y(t);r)dt+1√N∑rr√(Φ(Y(t);r)dW(r)(t), | (8) |
where the
A=[AvAw]=[˜bρ2vρw−˜dρv,˜s(1−ρv−ρw)−˜bρ2vρw−˜vρw,]. |
If we denote by
B=[BvvBvwBwvBww], |
where
Bvv=˜bρ2vρw+˜dρv,Bvw=Bwv=−˜bρ2vρw,Bww=˜bρ2vρw+˜s(1−ρv−ρw)+˜vρw, |
and factor it as
dY(t)=Adt+1√NgdW, | (9) |
where
We use the stochastic differential equation (9) to simulate the water-vegetation system and obtain averages of the expected time to desertification (see [10] or [13] for a quick or an extensive introduction respectively to the numerical solution of stochastic differential equations).
By identifying the parameters of the nondimensional deterministic (non-spatial) model in [12] with the mean field system obtained above (
Parameter | Definition | Estimated values | Units |
uptake rate of water | 1.5(trees) - 100(grass) | mm year |
|
yield of plant biomass | 0.002(trees) - 0.003(grass) | kg dry mass (mm) |
|
mortality rate | 0.18(trees) - 1.8(grass) | year |
|
precipitation | 250 - 750 | mm year |
|
evaporation rate | 4 | year |
|
Precipitation anomalies records, i.e. records of the deviations from a long term precipitation mean, have a negative trend in specific geographic drought events. In the state of California for instance, which experienced unusually long drought conditions, the (state) average of the precipitation anomalies for the past 16 years is
The results of the simulations are shown in Figure 3, portraying a roughly linear relationship between the time to desertification and the anomalies in precipitation in the range selected. The simulations were run using parameters for trees (top panel) and grass (low panel), with system capacity
A traditional approach for modeling interacting populations at the macroscopic level assumes that the terms in the equations that drive the dynamics represent the average effects of individual interactions in a general, all-inclusive way. Subsequent developments use those models as departing points for building theoretical extensions by incorporating further complexity to the equations, like the inclusion of spatial dependence by adding diffusion, or the introduction of "noise" terms. A different modeling approach is to start at the individual level, with explicit rules describing the interactions between individuals and their environment, [17]. This alternative implies the definition of a Markov jump process that constitutes the foundation for developing definitive model approximations that relate macro and microscopic dynamical levels. In this paper we have taken the latter approach for constructing a stochastic differential equation (continuous state space) that approximates the dynamics of an idealized water-vegetation system, initially conceived as a Markov jump process (with discrete states as a proxy for small scale). Thus, our work complements the existing literature on modeling noise in drylands, [21].
The diffusion approximation obtained, together with parameter values for vegetation and precipitation for semi-arid landscapes extracted from the literature, and data on decreasing precipitation trends in California, were used to estimate average times for desertification. For a fixed system capacity (
We remark that extended droughts may resemble desertification, but the return of seasonal precipitation events may recover the vegetation (see for instance [1] where desertification was limited to spatially localized areas). This suggests that the inclusion of patterns of precipitation anomalies restricted to relatively small areas would provide more reliable results. For systems with relatively small capacity we notice that the times to desertification may be reduced dramatically (see Figure 4).
Further work should also include long term variations of other climate related parameters. For instance, it has been documented that higher temperatures increase evapotranspiration rates [3], which have been observed over most of the United States, [8]. Another aspect that deserves attention is the inclusion of changes in the vegetation dynamics during dry periods, where vegetation mortality could be exacerbated. As is clearly pointed out in [28], neglecting the effects of intermittent precipitation on vegetation dynamics may influence the results considerably. Finally, the understanding of desertification will demand treatment with insightful stochastic space-time models.
The authors are grateful to C. Kribs and three anonymous reviewers for comments and suggestions that led to significant improvement of the paper.
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1. | E. M. Starodubtseva, I. N. Tsymbalov, D. A. Gorlova, K. A. Ivanov, A. B. Savel’ev, Polarization state control for high-peak-power applications, 2025, 42, 1084-7529, 285, 10.1364/JOSAA.544487 |
Event | Transition | Jump | Jump rate |
Vegetation biomass loss | |||
Incoming water | |||
Water evaporation | |||
Increase vegetation by | |||
water take up | |||
Parameter | Definition | Estimated values | Units |
uptake rate of water | 1.5(trees) - 100(grass) | mm year |
|
yield of plant biomass | 0.002(trees) - 0.003(grass) | kg dry mass (mm) |
|
mortality rate | 0.18(trees) - 1.8(grass) | year |
|
precipitation | 250 - 750 | mm year |
|
evaporation rate | 4 | year |
|
Event | Transition | Jump | Jump rate |
Vegetation biomass loss | |||
Incoming water | |||
Water evaporation | |||
Increase vegetation by | |||
water take up | |||
Parameter | Definition | Estimated values | Units |
uptake rate of water | 1.5(trees) - 100(grass) | mm year |
|
yield of plant biomass | 0.002(trees) - 0.003(grass) | kg dry mass (mm) |
|
mortality rate | 0.18(trees) - 1.8(grass) | year |
|
precipitation | 250 - 750 | mm year |
|
evaporation rate | 4 | year |
|