Something about the migration project.

# Poster

# 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 `$j$`

.

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