The absence of the first principles for biological systems in general, and in particular for vegetation populations where phenomena are interconnected makes their mathematical modelling complex. The theory of vegetation pattern formation rests on the self-organisation hypothesis and symmetry-breaking instability that provoke the fragmentation of the uniform cover. The symmetry-breaking instability takes place even if the environment is isotropic31,33,35. This instability may be an advection-induced transition that requires the pre-existence of the environment anisotropy due to the topography of the landscape34,39,40. Generally speaking, this transition requires at least two feedback mechanisms having a short-range activation and a long-range inhibition. In this respect, we consider three different vegetation models that are experimentally relevant systems: (i) the generic interaction redistribution model describing vegetation pattern formation which incorporates explicitly the facilitation, competition and seed dispersion nonlocal interactions (ii) the local nonvariational partial differential model described by a nonvariational Swift–Hohenberg type of model equation, and (iii) the reaction–diffusion system that incorporate explicetely water transport.
The interaction-redistribution approach
The integrodifferential model
This approach consists of considering a well-known logistic equation with nonlocal plant-to-plant interactions. Three types of interactions are considered: the facilitative (M_{f}(mathbf {r},t)), the competitive (M_{c}(mathbf {r},t)), and the seed dispersion (M_{d}(mathbf {r},t)) nonlocal interactions. To simplify further the mathematical modelling, we consider that the seed dispersion obeys a diffusive process (M_{d}(mathbf {r},t)approx nabla ^{2}b(mathbf {r},t)), with D the diffusion coefficient, b the biomass density, and (nabla ^{2}=partial ^2/partial x^2+partial ^2/partial y^2) is the Laplace operator acting in the (x,y) plane. The interaction-redistribution reads
$$begin{aligned} M_{i}=expleft{ frac{xi _{i}}{N_{i}}int b(mathbf {r}+mathbf {r}’,t)phi _i(r,t)dmathbf {r}’right} , { text{ with } } phi _i(r,t)= exp(-r/L_{i}) end{aligned}$$
(1)
where (i=f,c). (xi _i) represents the strength of the interaction, (N_i) is a normalisation constant. We assume that their Kernels (phi _i(r,t)) are exponential functions with (L_i) the range of their interactions. The facilitative interaction (M_{f}(mathbf {r},t)) favouring vegetation development. They involve the accumulation of nutrients in the neighbourhood of plants, the reciprocal sheltering of neighbouring plants against climatic harshness which improves the water budget in the soil. The range of the facilitative interaction (L_f) operates on the crown size. The competitive interaction operates over a length (L_c) and involves the below-ground structures, i.e., the rhizosphere. In nutrient-poor or/and in water-limited territories, lateral spreading may extend beyond the radius of the crown. This extension of roots relative to their crown size is necessary for the survival and the development of the plant in order to extract enough nutrients and/or water from the soil. When incorporating these nonlocal interactions in the paradigmatic logistic equation, the spatiotemporal evolution of the normalised biomass density (b(mathbf {r}, t)) in isotropic environmental conditions reads14
$$begin{aligned} partial _{t} b(mathbf {r},t)=b(mathbf {r},t)[1-b(mathbf {r},t)]M_{f}(mathbf {r},t)- mu b(mathbf {r},t)M_{c}(mathbf {r},t)+Dnabla ^{2}b(mathbf {r},t). end{aligned}$$
(2)
The normalisation is performed with respect to the total amount of biomass supported by the system. The first two terms in the logistic equation with nonlocal interaction Eq. (2) describe the biomass gains and losses, respectively. The third term models seed dispersion. The aridity parameter (mu) accounts for the biomass loss and gain ratio, which depends on water availability and nutrients soil distribution, topography, etc. The homogeneous cover solutions of Eq. (2) are: (b_{o}=0) which corresponds to the state totally devoid of vegetation, and the homogeneous cover solutions satisfy the equation
$$begin{aligned} mu =(1-b)exp (Delta b), end{aligned}$$
(3)
with (Delta =xi _{f}-xi _{c}) measures the community cooperativity if (Delta >0) or anti-cooperativity when (Delta <0). The bare state (b_{o}=0) is unstable (stable) (mu <1) ((mu >1)). The homogeneous cover state with higher biomass density is stable and the other is unstable. These solutions are connected by a saddle-node or a tipping point whose coordinates are given by (left{ b_{sn}=(Delta -1)/Delta ,mu _{sn}=e^{Delta -1}/Delta right}). The linear stability analysis of vegetated cover ((b_{s})) with respect to small fluctuations of the from (b(mathbf {r},t)=b_{s}+ delta b exp{sigma t+imathbf {k}cdot mathbf {r}}) with (delta b) small, yields the dispersion relation
$$begin{aligned} sigma (k)=left( b_{s}(1-b_{s})xi _{f}-b_{s}-frac{b_{s}(1-b_{s})xi _{c}}{(1+L_{c}^{2}k^{2})^{3/2}}right) e^{xi _{f}b_{s}}-Dk^{2}. end{aligned}$$
(4)
Given the spatial isotropy, the growth rate (sigma (k)) is a real quantity. This eigenvalue may become positive for a finite band of unstable modes which triggered the spontaneous amplification of spatial fluctuations towards the formation of periodic structures with a well-defined wavelength. At the symmetry-breaking instability the value of the critical wavenumber (k_c) marking the appearance of a band of unstable modes, and hence the symmetry-breaking instability, can be evaluated by two conditions: (sigma (k_c)=0) and (partial sigma /partial k|_{k_{c}}=0). These conditions yield the most unstable mode
$$begin{aligned} k_{c}^{2}=frac{1}{L_{c}^{2}}left[ left( frac{3b_{s}e^{xi _{f}b_{s}}(1-b_{s})xi _{c}L_{c}^{2}}{2D}right) ^{2/5}-1right] . end{aligned}$$
(5)
This critical wavenumber determines the wavelength of the periodic vegetation pattern (2pi /k_c) that emerges from the symmetry-breaking instability. Replacing (k_c) in the condition (sigma (k_{c})=0), we can then calculate the critical biomass density (b_{c}). The corresponding critical aridity parameter (mu _{c}) is provided explicitly by the homogeneous steady states Eq. (3).
Local model: a nonvariational Swift–Hohenberg model
The integrodifferential equation (2) can be reduced by means of a multiple-scale analysis to a simple partial differential equation, in the form of nonvariational Swift–Hohenberg equation. This reduction has been performed in the neighbourhood of the critical point associated with the nascent bistability14,32. The coordinates of the critical point are: the biomass density (b_c = 0), the cooperativity parameter (Delta _c=1), and the aridity parameter (mu _c=1). These coordinates are obtained from Eq. (3) by satisfying the double condition (partial mu /partial b_s=0) and (partial ^2mu /partial b_s^2=0). To apply a multiple-scale analysis it is necessary to define a small parameter that measures the distance from criticality and expand b, (mu), and (Delta) in the Taylor series around their critical values. The symmetry-breaking instability should be close to that critical point. To fulfil this condition, we must consider a small diffusion coefficient in order to include the symmetry-breaking instability in the description of the dynamics of the biomass density. This reduction is valid in the double limit of nascent bistability and close to the symmetry-breaking instability. In this double limit, the time-space evolution of biomass density obeys a non-variational Swift–Hohenberg model14
$$begin{aligned} partial _{t}u(mathbf {r},t)=-u(mathbf {r},t)(eta -kappa u(mathbf {r},t)+u(mathbf {r},t)^{2})+left[ nu -gamma u(mathbf {r},t)right] nabla ^{2}u(mathbf {r},t) -alpha u(mathbf {r},t)nabla ^{4}u(mathbf {r},t), vspace{0.3cm} end{aligned}$$
(6)
where (eta) and (kappa) are, respectively, the deviations of the aridity and cooperativity parameters from their values at the critical point. The linear and nonlinear diffusion coefficients (nu), (gamma), and (alpha) depend on the shape of kernels17. In addition to the bare state (u=0), the homogeneous covers obey
$$begin{aligned} u_{pm }=frac{kappa pm sqrt{kappa ^{2}-4eta }}{2}, end{aligned}$$
(7)
where the two homogeneous solutions (u_{pm }) are connected through the saddle-node bifurcation (left{ u_{sn}=kappa /2,eta _{sn}=kappa ^{2}/4right}), with (kappa >0). The solution (u_{-}) is always unstable even in the presence of small spatial fluctuations. The linear stability analysis of vegetated cover ((u_{+})) with respect to small spatial fluctuations, yields the dispersion relation
$$begin{aligned} sigma (k)=u_{+}(kappa -2u_{+})-(nu -gamma u_{+})k^{2}-alpha u_{+}k^{4}. end{aligned}$$
(8)
Imposing (partial sigma /partial k|_{k_{c}}=0) and (sigma (k_{c})=0), the critical mode can be determined
$$begin{aligned} k_{c}=sqrt{frac{gamma -nu /u_{c}}{2alpha }}, end{aligned}$$
(9)
where (u_{c}) satisfies (4alpha u_{c}^2(2u_{c}-kappa )=(2gamma u_{c}-nu )^2). The corresponding aridity parameter (eta _{c}) can be calculated from Eq. (7).
The reaction–diffusion approach
The second approach explicitly adds the water transport by below ground diffusion. The coupling between the water dynamics and the plant biomass involves positive feedbacks that tend to enhance water availability. Negative feedbacks allow for an increase in water consumption caused by vegetation growth, which inhibits further biomass growth.
The modelling considers the coupled evolution of biomass density (b(mathbf {r},t)) and groundwater density (w(mathbf {r},t)). In its dimensionless form, this model reads33
$$begin{aligned} frac{partial b}{partial t}= & {} frac{gamma w}{1+omega w}b-b^{2}-theta b+nabla ^{2}b, end{aligned}$$
(10)
$$begin{aligned} frac{partial w}{partial t}= & {} p-(1-rho b)w-w^{2}b+delta nabla ^{2}(w-beta b). end{aligned}$$
(11)
The first term in the first equation describes plant growth at a constant rate ((gamma /omega)) that grows linearly with w for dry soil. The quadratic nonlinearity (-b^{2}) accounts for saturation imposed by poor nutrients soil. The term proportional to (theta) accounts for mortality, grazing or herbivores. The mechanisms of dispersion are modelled by a simple diffusion process. The groundwater evolves due to a precipitation input p. The term ((1-rho b)w) in the second equation accounts for the evaporation and drainage, that decreases with the presence of vegetation. The term (w^{2}b) models the water uptake by the plants due to the transpiration process. The groundwater movement follows the Darcy’s law in unsaturated conditions; that is, the water flux is proportional to the gradient of the water matric potential41. The matric potential is equal to w, under the assumption that the hydraulic diffusivity is constant41. To model the suction of water by the roots, a correction to the matric potential is included; (-beta b), where (beta) is the strength of the suction.
Source: Ecology - nature.com
