Competition in a nonlocal nonlinear death term

Figure 3.6:Perturbation growth rate of the most unstable mode as a func-tion of the competifunc-tion strength for different values of the exponents.s=1 full line,s=2 dashed line, ands=3 dotted-dashed line. Other parameters:

α=1,β0=5, andR=8.

Similar results can be obtained for different growth rates, for example con-sidering a family of stretched exponentials in the probability of overcoming long-range competitionr( ˜ρ, δ)=exp −δρ˜^s

. This gives a perturbation growth rate of the form

λ(k)=−αρ0

1

1−ρ0 +δsρ^p−1₀ G(k)ˆ

!

, (3.10)

where the stationary density,ρ0 has to be obtained numerically from Eq. (3.4).

The value of the competition parameter at the transition to patternsδcalso has to be obtained numerically for a given set of values ofs,α, andβ0. This critical value is shown in Fig. 3.6 using a top-hat kernel for different values ofs. It is represented the value of the perturbation growth rate of the most unstable mode λ(kc) as a function of the competition strengthδ.

This further confirms our result that competition is the only necessary ingredient for the formation of vegetation patterns in the present framework, and that this does not depend on the functional form of the probability of surviving competition (growth rate) provided it verifies the requirements given by Eq. (3.3).

3.3

Competition in a nonlocal nonlinear death term

In the previous section we have considered that plant death occurs at a constant rate and the birth process takes place following a sequence of seed production, dispersal and establishment. The establishment of new seeds has been studied as a process depending on the density of vegetation in a given neighborhood.

However, it is possible to consider it affecting the death rate.

In this framework, the population growth has two stages, namely seed produc-tion at rateβ0and local seed dispersal. This means that once a seed is produced by a plant, it still has to compete for available space, limiting the density of

vegetation to a maximum valueρmax. As it was done before, we choose units so thatρmax = 1. On the other hand, the death of plants is influenced by the availability of resources and thus include nonlocal interactions. When water is abundant the competition for it is not relevant and plants die at a typical constant rateα0. However, the scarcity of resources promotes the death of vegetation, and this is included in the model by an additional factorhin the death term, the probability of dying because of competition, 0≤h≤1. That is, the probability of not being able to overcome competitionh( ˜ρ, δ) = 1−r( ˜ρ, δ), whereris the function introduced in Sec. 3.2. The probability of dying because of competition for resources has to increase with the density of vegetation

∂h( ˜ρ, δ)

∂ρ˜ ≥0, (3.11)

and it must tend to its maximum value in the limit of extremely arid systems (δ → ∞) and to vanish when water is not a constraint for vegetation growth (δ →0). These properties can be derived from the properties of the functionr and its relationship withh.

Under these conditions, the model equation for the density of vegetation is

∂ρ(x,t)

∂t =βρ(x,t)(1−ρ(x,t))−α0h( ˜ρ(x,t), δ)ρ(x,t), (3.12) where ˜ρ(x,t) is the nonlocal density of vegetation atx, ˜ρ(x,t) = R

ρ(x⁰,t)G(|x− x⁰|)dx⁰, and G(x) is the kernel function that defines an interaction range and modulates its strength with the distance from the focal plant, as it was in Sec. 3.2.

Also following the step of this previous section, we study the existence of patterns in this model.

First of all, the stationary solutions of Eq. (3.12),ρ0, are obtained solving βρ0(1−ρ0)−α0h(ρ0, δ)ρ0=0, (3.13) that has a trivial solution,ρ0=0 referring to the desert state. The vegetated state must be obtained from

β(1−ρ0)−α0h(ρ0, δ)=0, (3.14) once the functionhhas been chosen.

Second of all, the formation of patterns in the system has to be studied through a linear stability analysis of Eq. (3.12), introducing a small perturbation to the sta-tionary homogeneous state,ρ0. Considering thatρ(x,t)=ρ0+ψ(x,t), with1, the perturbation evolves according to the following linear integro-differential equation that can be solved using the Fourier transform to obtain the growth rate of the perturbation,

λ(k)=β(1−2ρ0)−α0h(ρ0, δ)−α0ρ0h⁰(ρ0, δ) ˆG(k), (3.16)

3.3. COMPETITION IN A NONLOCAL NONLINEAR DEATH TERM

Figure 3.7: Probability of dying because of long range competition as a function of the density of vegetation. Left panel,h( ˜ρ(x,t), δ) =δρ. Right˜ panel,h( ˜ρ(x,t), δ)= _δ+^δρ˜_ρ˜. From top to bottomδ=1, δ=0.8, δ=0.6, δ=0.4

in both panels.

where, again, ˆG(k) is the Fourier transform of the Kernel function. Using Eq. (3.13), one finally gets

λ(k)=−ρ0

hβ+α0h⁰(ρ0, δ) ˆG(k)i

, (3.17)

that shows that, as it was in the case of models with nonlocal interactions in the birth rate, that the patterns appear only when the Fourier transform of the kernel takes negative values. To investigate the particular behavior of this model in a simple situation, one needs to choose a functionh.

The simplest case that allows a complete analysis is to choose a probability of dying because of competition growing linearly with the nonlocal density of vegetation,

h( ˜ρ(x,t)=δρ(x,˜ t). (3.18) The behavior of this functionhis shown in the left panel of Fig. 3.7 for different values ofδ. The choice of this functional form inhrestricts the domain of the parameterδ. It has to beδ∈[0,1] since it must beh≤1 to represent a probability.

The nontrivial stationary solution, Eq. (3.14), is ρ0= β

β+α0δ. (3.19)

This solution has the proper behavior in the limitsδ→0 andδ→ ∞. The growth rate of the perturbation is

λ(k)=− β β+α0δ

hβ+α0δG(k)ˆ i

, (3.20)

from where we obtain a transition to pattern at, δc=− β

α0G(kˆ c), (3.21)

Figure 3.8:Distribution of vegetation of the model in a patch of size 100× 100 with a linear probability of dying because of long-range competition,

Eq. (3.18). (a)δ=0.7, (b)δ=0.8, (c)δ=0.9.α0=1,β=0.1,R=8.

where kc is the most unstable mode, i.e., that with the highest growing rate.

Eq. (3.21) limits the possible value of the birth and the death rates for a given kernel. They have to take values such asδc ≤1. This is a new condition that was not present in the model presented in Sec. 3.2, and appears because of the constraints imposed on the values of the competition parameter by the linear functionh. Forα0 = 1, β= 0.1 and a top-hat kernel of radiusR = 8, patterns emerge for a competition strengthδc ≈0.747. As it can be observed in Fig. 3.8, the homogeneous distribution is the stationary solution whenδ < δc(Panel a), while patterns (stripes and spots) appear otherwise (Panels b and c respectively).

A different function for the probability of dying might be chosen¹,

h( ˜ρ(x,t), δ)= δρ˜

δ+ρ˜, (3.22)

that is shown in the right panel of Fig. 3.7 for different values ofδ. Although this makes the analysis much more complicated, it allows the competition strength parameterδto take any positive real value. We present some results showing that the functional form ofhis not relevant for pattern formation.

The nontrivial stationary solution is

ρ0 =−[(α0+β)δ−β]+p

[(α0+β)δ−β]²+4β²δ

2β , (3.23)

1It is important to justify thatδhas no dimension, and there is a saturating density,κ, that we have set toκ=1 so thatr=_δκ+^δρ˜_ρ˜.