Generalized Cahn-Hilliard equation for biological applications

Khain, Evgeniy; Sander, Leonard M.

doi:10.1103/physreve.77.051129

Cited by 102 publications

(65 citation statements)

References 41 publications

(41 reference statements)

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“…The results of our calculations together with those of Monte Carlo simulations are discussed below. Let us also note that for some biologically inspired lattice models (i.e., the wound-healing problem) other approximate descriptions are possible based, for example, on the Cahn-Hilliard equation [40].…”

Section: Mean-field Approximationmentioning

confidence: 99%

Coexistence and critical behavior in a lattice model of competing species

Wendykier¹,

Lipowski²,

Ferreira³

2011

Phys. Rev. E

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In the present paper we study a lattice model of two species competing for the same resources. Monte Carlo simulations for d = 1,2, and 3 show that when resources are easily available both species coexist. However, when the supply of resources is on an intermediate level, the species with slower metabolism becomes extinct. On the other hand, when resources are scarce it is the species with faster metabolism that becomes extinct. The range of coexistence of the two species increases with dimension. We suggest that our model might describe some aspects of the competition between normal and tumor cells. With such an interpretation, examples of tumor remission, recurrence, and different morphologies are presented. In the d = 1 and d = 2 models, we analyze the nature of phase transitions: they are either discontinuous or belong to the directed-percolation universality class, and in some cases they have an active subcritical phase. In the d = 2 case, one of the transitions seems to be characterized by critical exponents that differ from directed-percolation ones, but this transition could be also weakly discontinuous. In the d = 3 version, Monte Carlo simulations are in a good agreement with the solution of the mean-field approximation. This approximation predicts that oscillatory behavior occurs in the present model but only for d 2. For d 2, a steady state depends on the initial configuration in some cases.

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Section: Mean-field Approximationmentioning

confidence: 99%

Coexistence and critical behavior in a lattice model of competing species

Wendykier¹,

Lipowski²,

Ferreira³

2011

Phys. Rev. E

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“…They identified that density-dependent rates of dispersal can lead to separation of a mixed fluid into two phases that are separated in distinct spatial regions, subsequently leading to pattern formation. The principle of density-dependent dispersal, switching between dispersion and aggregation as local density increases, has become a central mathematical explanation for phase separation in many fields (21) such as multiphase fluid flow (22), mineral exsolution and growth (23), and biological applications (24)(25)(26)(27)(28). Although aggregation due to individual motion is a commonly observed phenomenon within ecology, application of the principles of phase separation to explain pattern formation in ecological systems is absent both in terms of theory and experiments (25,26).…”

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confidence: 99%

Phase separation explains a new class of self-organized spatial patterns in ecological systems

Liu

Doelman

Rottschäfer

et al. 2013

Proc. Natl. Acad. Sci. U.S.A.

153

156

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The origin of regular spatial patterns in ecological systems has long fascinated researchers. Turing's activator-inhibitor principle is considered the central paradigm to explain such patterns. According to this principle, local activation combined with longrange inhibition of growth and survival is an essential prerequisite for pattern formation. Here, we show that the physical principle of phase separation, solely based on density-dependent movement by organisms, represents an alternative class of self-organized pattern formation in ecology. Using experiments with self-organizing mussel beds, we derive an empirical relation between the speed of animal movement and local animal density. By incorporating this relation in a partial differential equation, we demonstrate that this model corresponds mathematically to the wellknown Cahn-Hilliard equation for phase separation in physics. Finally, we show that the predicted patterns match those found both in field observations and in our experiments. Our results reveal a principle for ecological self-organization, where phase separation rather than activation and inhibition processes drives spatial pattern formation.he activator-inhibitor principle, originally conceived by Turing in 1952 (1), provides a potential theoretical mechanism for the occurrence of regular patterns in biology (2-6) and chemistry (7-9), although experimental evidence in particular for biological systems has remained scarce (3, 4, 10). In the past decades, this principle has been applied to a wide range of ecological systems, including arid bush lands (11-15), mussel beds (16, 17), and boreal peat lands (18,19). The principle, in which a local positive activating feedback interacts with large-scale inhibitory feedback to generate spatial differentiation in growth, birth, mortality, respiration, or decay, explains the spontaneous emergence of regular spatial patterns in ecosystems even under near-homogeneous starting conditions. Physical theory offers an alternative mechanism for pattern formation, proposed by Cahn and Hilliard in 1958 (20). They identified that density-dependent rates of dispersal can lead to separation of a mixed fluid into two phases that are separated in distinct spatial regions, subsequently leading to pattern formation. The principle of density-dependent dispersal, switching between dispersion and aggregation as local density increases, has become a central mathematical explanation for phase separation in many fields (21) such as multiphase fluid flow (22), mineral exsolution and growth (23), and biological applications (24-28). Although aggregation due to individual motion is a commonly observed phenomenon within ecology, application of the principles of phase separation to explain pattern formation in ecological systems is absent both in terms of theory and experiments (25,26).Here, we apply the concept of phase separation to the formation of spatial patterns in the distribution of aggregating mussels. On intertidal flats, establishing mussel beds exhibit spatial self-org...

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“…We can mention, for instance, population dynamics (see [12]), bacterial films (see [22]), wound healing and tumor growth (see [10], [21] and [29]), thin films (see [38] and [39]), image processing and inpainting (see [2], [3], [4], [7], [8] and [14]) and even the rings of Saturn (see [40]) and the clustering of mussels (see [25]). …”

Section: Introductionmentioning

confidence: 99%