Theory of neutral clustering for growing populations

Bahram Houchmandzadeh

Phys. Rev. E 80, 051920 – Published 24 November 2009

Abstract

The spatial distribution of most species in nature is nonuniform. We have shown recently [B. Houchmandzadeh, Phys. Rev. Lett. 101, 078103 (2008)] on an experimental ecological community of amoeba that the most basic facts of life—birth and death—are enough to cause considerable aggregation which cannot be smoothened by random movements of the organisms. This clustering, termed neutral and always present, is independent of external causes and social interaction. We develop here the theoretical groundwork of this phenomenon by explicitly computing the pair-correlation function and the variance to mean ratio of the above neutral model and its comparison to numerical simulations.

Received 30 July 2009

DOI:https://doi.org/10.1103/PhysRevE.80.051920

Authors & Affiliations

Bahram Houchmandzadeh

Laboratoire de Spectrométrie Physique, CNRS & Grenoble Universités, Boîte Postale 87, 38402 St. Martin d’Hères Cedex, France

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 80, Iss. 5 — November 2009

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Examples of (a) random uniform distribution; (b) patchy nonuniform distribution. A robust quantifier for patchiness is computed by dividing the space into contiguous squares (quadrats) of size $L$ , counting the number $N_{i}$ of particles in each quadrat and forming the ratio of the variance to the mean $(V_{m})$ of these numbers. If the $V_{m} \approx 1$ the distribution is uniform or (Poisson) random (a). If $V_{m} > 1$ , the distribution is nonuniform or clustered [(b), $V_{m} \approx 6$ computed over the squares shown in panel (a)]. $V_{m} < 1$ denotes an ordered distribution and is seldom encountered in natural environments.Reuse & Permissions
Figure 2
Example of spatial distribution in an experimental community [15]: map of $\approx 55 000$ pdsA-D. Discoïdum mutant after 72 h (7.5 generations) of growth in a petri dish. The map is obtained by scanning the bottom of the petri dish by an automated 10X objective and taking contiguous photograph; the rectangle denotes the relative size of one photograph. The position of each amoeba in each photograph is detected by an image analysis program. The result is assembled into the above map, where each dot represents one amoebae (see the Appendix). $V_{m} = 35$ for quadrats the size of the black rectangle. Identical results are obtained by using WT D. Discoïdum.Reuse & Permissions
Figure 3
(Color online) (a) A stochastic numerical simulation of the number of individuals $n_{i} (t)$ as of function of generation time $(α t)$ for 1000 wells. Each curves represents the population in one well. To generate the paths, at each time step, the number of birth and death are generated by a Poisson process. $α = 1$ . $μ = 0.9$ . [(a), inset] evolution of the average $⟨ n (t) ⟩$ (black circles) and $\sqrt{V (t)}$ (red squares) and their comparison to theoretical expressions (3, 4). (b) A visual map of the population number in each well after 60 generations. Wells are arranged into a $25 \times 40$ array, each dot represents 50 individuals.Reuse & Permissions
Figure 4
(Color online) Numerical simulation of a growing population and its pair-correlation function. Data for numerical simulations are similar to the experimental ecosystems of [15]: $α = 1$ , $μ = 0$ , $D = 1 / 2$ , and $c_{0} = 9.6 \times 10^{- 2}$ ; time scales are in generation units, spatial scales in migration length units. (a) Spatial map of the organism at $t = 6$ ; each dot represent one organism. (b)–(e) pair-correlation functions for four times: (b) $t = 0.2$ ; (c) $t = 1$ ; (d) $t = 3$ ; and (e) $t = 6$ . Circles and shaded area: average and $\pm$ one standard deviation computed on 25 numerical simulations; (red online) solid lines: theoretical predictions. The small difference at short distances for large times (e) is due to the finite size of the ecosystem.Reuse & Permissions
Figure 5
(Color online) The theoretical $V_{m}$ (solid curves) and its comparison to the function computed on individual-based numerical simulations (symbols). Data represent averages over 25 simulated populations with the same parameters as in Fig. 4, and error bars are $\pm$ one standard deviation over the same samples. (a) $V_{m}$ as a function of time $t$ for three different values of $L : L = 0.4$ (circles); $L = 1$ (diamonds); and $L = 2.5$ (squares). (b) $V_{m}$ as a function of scale $L$ for three different values of time $t : t = 0.2$ (circles); $t = 1$ (diamonds); and $t = 6$ (squares). Thick solid bars on the left side represent saturation values given by expression (14).Reuse & Permissions
Figure 6
(Color online) The function $h (u)$ (solid curve) and $h (u) / \sqrt{π u / 2}$ (dotted curve). Inset: comparison of the function $V_{m} (1, t)$ (circles) to its long-time approximation $exp (t) / π t$ (solid curve); $d = 2$ and $μ = 0$ .Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics