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Patchiness and Demographic Noise in Three Ecological Examples

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Abstract

Understanding the causes and effects of spatial aggregation is one of the most fundamental problems in ecology. Aggregation is an emergent phenomenon arising from the interactions between the individuals of the population, able to sense only—at most—local densities of their cohorts. Thus, taking into account the individual-level interactions and fluctuations is essential to reach a correct description of the population. Classic deterministic equations are suitable to describe some aspects of the population, but leave out features related to the stochasticity inherent to the discreteness of the individuals. Stochastic equations for the population do account for these fluctuation-generated effects by means of demographic noise terms but, owing to their complexity, they can be difficult (or, at times, impossible) to deal with. Even when they can be written in a simple form, they are still difficult to numerically integrate due to the presence of the “square-root” intrinsic noise. In this paper, we discuss a simple way to add the effect of demographic stochasticity to three classic, deterministic ecological examples where aggregation plays an important role. We study the resulting equations using a recently-introduced integration scheme especially devised to integrate numerically stochastic equations with demographic noise. Aimed at scrutinizing the ability of these stochastic examples to show aggregation, we find that the three systems not only show patchy configurations, but also undergo a phase transition belonging to the directed percolation universality class.

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Notes

  1. Even worse, in some cases, the described derivation of continuous approaches relying on standard field-theoretical techniques does not produce any Langevin equation, i.e. the resulting generating functional cannot be cast into any standard stochastic equation.

  2. In the assumption of D and D 2 being constant, it is implicitly assumed that there is no intraspecific variability or other kind of heterogeneity in the distribution of these dispersal rates.

  3. Of course, active aggregation must be added on top of this to bring the patterns to a more realistic scenario. However, our objective is to show that there may be two complementary origins of pattern that must be integrated, one arising from stochastic demographic fluctuations and the other owing to active aggregation.

  4. Note that, in this case, we are using as a reference at the microscopic level the real interactions and patterns observed in Nature (see [17, 18, 36] for some illustrative examples).

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Acknowledgements

We gratefully acknowledge support from the Cooperative Institute for Climate Science (CICS) of Princeton University and the National Oceanographic and Atmospheric Administration’s (NOAA) Geophysical Fluid Dynamics Laboratory (GFDL), the National Science Foundation (NSF) under grant OCE-1046001, the Spanish MICINN-FEDER under project FIS2009-08451, and from Junta de Andalucía (Proyecto de Excelencia P09FQM-4682). MAM thanks I. Dornic, H. Chaté and, C. López, and F. Ramos, for a long term collaboration on some of the issues presented here.

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Authors and Affiliations

  1. Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ, 08544-1003, USA

    Juan A. Bonachela & Simon A. Levin

  2. Instituto Carlos I de Física Teórica y Computacional, Facultad de Ciencias, Universidad de Granada, 18071, Granada, Spain

    Miguel A. Muñoz

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  1. Juan A. Bonachela
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Correspondence to Juan A. Bonachela.

Appendix

Appendix

We have numerically shown that the critical exponents measured in all of the examples discussed in this paper are directed-percolation like. Here we justify that result by analyzing their corresponding Langevin equations Eqs. (3), Eqs. (5), and Eq. (7).

As conjectured years ago by Janssen and Grassberger [44, 45] all systems exhibiting a phase transition into a unique absorbing state, with a single-component order parameter, and no extra symmetry or conservation law, belong to the same universality class whose most distinguished representative is directed percolation (DP). In a field theoretical description this universality class is represented by the Reggeon Field theory (RFT), [44, 45] which in terms of a Langevin equation reads:

$$ {\displaystyle{\partial\rho(\mathbf {x},t) \over\partial t}} = \nabla^2 \rho (\mathbf {x},t) + a \rho(\mathbf {x},t) -b \rho^2(\mathbf {x},t) + \sqrt{\rho(\mathbf {x},t)} \eta(\mathbf {x},t) , $$
(9)

where ρ(x,t) is the density field at position x and time t, a and b are parameters, and η(x,t) is a delta-correlated Gaussian noise, 〈η(x,t)η(x′,t′)〉=Dδ(x−x′)δ(t−t′).

The previous conjecture was confirmed in a large number of computer simulations, series expansion analysis, field theoretical studies, etc. Indeed, the DP universality class has proved to be extremely robust against the modification of many details in the microscopic models. The conjecture of universality has been extended systems with an infinite number of absorbing states [46, 47] (see also [48]). Also, Grinstein et al. extended the conjecture to the case of multicomponent systems [49] (see also [50]). Considering, for the sake of simplicity a two-component system with absorbing states, it can generically be described by a set of Langevin equations whose linearized dynamics (ignoring the Laplacian terms and the noise) is generically:

$$ \begin{array}{ll} \partial_{t}\rho_1(\mathbf {x},t)=a_{1,1}\rho_1+ a_{1,2}\rho_2 \cr\noalign{\vspace{3pt}} \partial_{t}\rho_2(\mathbf {x},t)= a_{2,1}\rho_1 + a_{2,2}\rho_2 \end{array} $$
(10)

and diagonalizing the matrix of the linear coefficients:

$$ \begin{array}{@{}ll} \partial_{t}\xi_1(\mathbf {x},t)=\lambda_1\xi_1 \cr\noalign{\vspace{3pt}} \partial_{t}\xi_2(\mathbf {x},t)= \lambda_{2}\xi_2 \end{array} $$
(11)

where λ 1 and λ 2 are the associated eigenvalues and ξ 1 and ξ 2 are the corresponding eigenvectors. The existence of the absorbing state implies that the two eigenvalues are negative. At the critical point one of the two eigenvalues, say λ 1 vanishes. If the other one does not, it remains negative, and then fluctuations in ξ 2 continue to decay exponentially with time. This implies that ξ 2 does not experience critical fluctuations in the vicinity of the transition, and so can be integrated out of the problem. Hence, asymptotically, the set of Langevin equations can be reduced to single relevant equation which is precisely Eq. (9), ensuing DP behavior.

On the other hand, if the secondary field is conserved (which, in particular, enforces a 2,1 and a 2,2 to vanish) or there is a hidden symmetry imposing λ 1 and λ 2 to vanish at the same point, this conclusion can break down. In both of these cases, at the system critical point, the two fields are simultaneously critical (or “massless” or “gapless” using the field theory jargon).

Systems in which the density field is coupled to a secondary conserved field indeed exhibit non-DP critical behavior. This is the case of, for instance, (i) systems related to self-organized criticality in which the activity field is coupled to a conserved non-diffusive background field [51–53] and (ii) systems in which the secondary field is conserved but diffusive [54, 55].

Let us now return to the three models discussed in this paper.

  1. (a)

    Let us first scrutinize the simpler case of vegetation patterns as described by Eqs. (5). At the critical point the expectation value of ρ vanishes, while ϕ converges on average to a value R/κ and, for generic values of ρ, it converges to ϕ st =(R−w 2 ρ)/κ. Let us remark that these are the noiseless or mean-field expectation values. Defining a new field ψ=ϕ−ϕ st and linearizing the dynamics as above, it is straightforward to see that this is an instance of the non-symmetric case discussed above in which the linear terms in both equations do not vanish simultaneously: DP behavior is predicted. Observe also that, after shifting the secondary field, a standard saturation term proportional to −ρ 2 appears in the first equation, i.e. the competition for the available resource (water) acts as an inhibitor generating the necessary effective carrying capacity term (see main text).

  2. (b)

    The case of the stochastic Levin-Segel model, as defined by Eqs. (3) is a bit trickier. Observe that on average one expects (at a noiseless or mean-field level) ϕ=ρw 2/λ, which implies that when ρ vanishes so does ϕ opening the door to non-DP scaling. However, closer inspection reveals that the noise in the equation for ϕ is a higher order one as it is proportional to \(\sqrt{\rho\phi}\). It is therefore irrelevant in the renormalization group sense around the DP-like fixed point (indeed, we have verified numerically that the critical exponents of the modified version of Eqs. (3) without such a term coincide, within error bars, with those of the original ones—results not shown). In this way, the equation for ∂ t ϕ becomes asymptotically a deterministic one. Then, ϕ(x) can be written as a function of ρ(x,t), i.e. ϕ(x,t)=F(ρ(x,t)) where F is an unspecified function which can be formally obtained by integration of the equation for ∂ t ϕ. Expanding such a function in power series of ρ, plugging in the result in the equation for ∂ t ρ and neglecting higher-order terms, one readily recovers Eq. (9), and hence DP scaling. The field ϕ is just a slave mode of ρ (proportional to it up to leading order), inheriting all its critical properties from it.

  3. (c)

    Finally, for the non-local Langevin model for Brownian bugs, it is easy to see that by performing a coarse graining in which scales below l are scaled-up, one recovers effectively Eq. (9). The main point is that nonlocal but finite-range interaction becomes local in the renormalization group perspective once a sufficient amount of coarse graining has been considered. See [23] for further details.

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Bonachela, J.A., Muñoz, M.A. & Levin, S.A. Patchiness and Demographic Noise in Three Ecological Examples. J Stat Phys 148, 724–740 (2012). https://doi.org/10.1007/s10955-012-0506-x

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