Starvation Driven Diffusion as a Survival Strategy of Biological Organisms

Abstract

The purpose of this article is to introduce a diffusion model for biological organisms that increase their motility when food or other resource is insufficient. It is shown in this paper that Fick’s diffusion law does not explain such a starvation driven diffusion correctly. The diffusion model for nonuniform Brownian motion in Kim (Einstein’s random walk and thermal diffusion, preprint http://amath.kaist.ac.kr/papers/Kim/31.pdf, 2013) is employed in this paper and a Fokker–Planck type diffusion law is obtained. Lotka–Volterra type competition systems with spatial heterogeneity are tested, where one species follows the starvation driven diffusion and the other follows the linear diffusion. In heterogeneous environments, the starvation driven diffusion turns out to be a better survival strategy than the linear one. Various issues such as the global asymptotic stability, convergence to an ideal free distribution, the extinction and coexistence of competing species are discussed.

Appendix: Derivation of Non-isothermal Diffusion

Let $\{x^{i}:i\in\mathbb{Z}\}$ be a mesh grid and x ^i+1/2:=(x ⁱ+x ⁱ⁺¹)/2 be the middle point between two adjacent grid points. Let V(x ⁱ) be the number of particles at x ⁱ, which jump randomly to one of two adjacent grid points every time interval Δt(x ⁱ). Define the walk length by Δx(x ^i+1/2):=x ⁱ⁺¹−x ⁱ and let Δx(x ⁱ):=x ^i+1/2−x ^i−1/2.

Then, the particle flux that crosses the a midpoint x ^i+1/2 from left to right is ${V\over2\varDelta t} |_{x=x^{i}}$ and the one from right to left is ${V\over2\varDelta t} |_{x=x^{i+1}}$. Notice that the particle density is given by $v={V\over\varDelta x}$, and hence the net flux is

$$ \mathbf{f}\bigl(x^{i+{1\over2}}\bigr)= {\varDelta x\,v\over2\varDelta t} \bigg|_{x^{i}}-{\varDelta x \,v\over2\varDelta t} \bigg|_{x^{i+1}} =-{\varDelta x|_{x^{i+{1\over2}}}\over2} \biggl({{\varDelta x\over \varDelta t}v |_{x^{i}}-{\varDelta x\over\varDelta t}v |_{x^{i+1}}\over x^i-x^{i+1}} \biggr). $$

(41)

If the Brownian motion or the random walk is in a homogeneous environment, we may assume that the mean free path Δx and the collision time Δt are constant, and hence ${\varDelta x\over \varDelta t}$ in the parenthesis of (41) can be taken out. However, if the temperature is not spatially constant, for example, they depend on the space variable and should stay inside. In conclusion, the diffusion flux for a non-isothermal case in n space dimensions is given by

$$ \mathbf{f}= -{D\over S}\nabla (S v ),\quad D:= {|\varDelta x|^2\over 2n\varDelta t},\ S:={\varDelta x\over\varDelta t}, $$

(42)

where S and D are called the walk speed and diffusivity, respectively. Notice that the walk length is given by |Δx|=2nD/S. Therefore, the corresponding non-isothermal diffusion equation is

$$(3)\quad v_t=\nabla\cdot \biggl( {D\over S}\nabla (S v ) \biggr). $$

One might numerically check that this diffusion model gives the correct behavior of nonuniform random walks with nonconstant Δx and Δt. The numerical solutions of this diffusion model have been compared with Monte Carlo simulations in Kim (2013).