My caption 😄

# Migration and Microbial Community

Something about the migration project.

# Consumer-resource model

Here I use MacArthur’s consumer-resource model to simulate the microbe growing in a chemostat. The model discription is as below. Other details are extended based on this model.

## Model description

I modify MacArthur’s consumer resource model to include consumers with stoichiometric metabolism. The model consists of one supplied resources $R_1$ exploited by a specialist $X_1$ and a generalist consumers $X_2$, while the specialist consumes supplied resources and secretes another resource $R_2$ as by-product.

Given that total number of resource type $P$ and total number of consumers $S$ in this system, the growth rate of consumer $i$ is defined by

$$\frac{dX_i}{dt}=\sum^P_{j=1} X_iw_j^iu_j^i-m_iX_i$$

where $X_i$ and $R_j$ are the biomass of consumer $i$ and resource $j$, respectively. $m_i$ is the density-independent loss rate of consumers $j$, $w_{ij}$ is the conversion efficiency of resource $j$ converted into biomass of consumer $i$, and $u_{ij}$ denotes the uptaking rate for consumer $i$ to use resource $j$.

Resource is supplied in chemostat fashion. Let the dynamics of resource $j$ be given by

$$\frac{dR_j}{dt}=J(R_{supply,j}-R_j)+\sum_{i=1}^{S}\sum_{k=1}^{P} D_{kj}^i u_{k}^{i} R_k X_i$$

The first term defines the resource supplement from environment external to this system, while the second term denote that from cross-feeding. In the first term, resource is supplied at flow rate $J$ and inflow concentration $R_{supply,j}$. In the second term, $D_{jk}^i$ is a stoichiometric matrix of consumer that describes the metabolic networks of consumer $i$ which uptakes and transforms resource $k$ into secreted resource $j$.

Thanks to the helpful information from this post for editing equation in markdown files.