# rescomp: An R package for efficient ODE modelling of resource competition

#### Christopher R. P. Brown, Jan Engelstaedter, and Andrew D. Letten

rescomp is an R package for generating, simulating and visualising ODE models of consumer-resource interactions. In essence, it is a consumer-resource modelling focused interface to the canonical deSolve package.

# Model formulation

All `rescomp` models take the general form,

\[ \frac{dN_{i}}{dt} = N_{i}(\mu_{i}(R_{1},R_{2},...)-m) \,, \]

\[ \frac{dR_{j}}{dt} = \Psi_{j} (R_{j}) - \sum_{i = 1}^{n}Q_{ij} \mu_{ij}(R_j) N_i \,, \]

where \(N_{i}\) is the population density of consumer \(i\), \(R_{j}\) is the density/concentration of resource \(j\), \(\mu_{i}()\) is the per capita consumer functional response of consumer \(i\), \(m\) is the per capita mortality rate (constant or intermittent), \(\Psi_{j}(R_j)\) is the resource supply function, and \(Q_{ij}\) is the resource quota of consumer \(i\) on resource \(j\) (the amount of resource per unit consumer).

From this general form, different model formulations are distinguished by: i) the number of consumers/resources; ii) the form of the consumer functional response; iii) the mode of resource supply; iv) the type of resource; and v) any non-autonomous behaviour including time dependent model parameters and/or instantaneous changes in consumer/resource density (e.g. in batch transfer).

## Consumer equations

The consumer growth function can take one of three forms:

1. linear (type I),

\[ \mu_{ij}(R_{j}) = a_{ij}R_{j} \,, \]

where \(a_{ij}\) is a resource consumption constant.

2. nonlinear type II (aka Monod)

\[ \mu_{ij}(R_{j}) = \mu_{max_{ij}}\frac{R_{j}}{k_{ij} + R_{j}} \,, \]

where \(\mu_{max_{ij}}\) is the maximum growth rate and \(k_{ij}\) is the half saturation constant for consumer \(i\) on resource \(j\).

3. nonlinear type III (sigmoidal),

\[ \mu_{ij}(R_{j}) = \mu _{max_{ij}}\frac{R_{j}^n}{k_{ij}^n + R_{j}^n} \, , \qquad n > 1 \,. \]

An alternative parameterisation to all three functional responses is to incorporate a consumption efficiency term, \(c_{ij}\). The type I model becomes \(\mu_{ij}(R_{j}) = a_{ij}c_{ij}R_{j}\), and for type II and III, \(a_{ij}c_{ij}\) is substituted for \(\mu _{max_{ij}}\). NB. this functional form is mutually exclusive with the specification of resources quotas in the resource equations, i.e. the \(Q_{ij}\)s are dropped from the resource equations.

## Resource equations

The resource supply function, \(\Psi_{j}(R_j)\), can also take several forms. For instance, the resources can be biological and grow logistically,

\[ \Psi_{j} (R_{j}) = r_{j}R_{j}\left(1-\frac{R_{j}}{K_{j}}\right) \,, \]

where \(r_j\) is the resource intrinsic rate of increase and \(K_j\) is the resource carrying capacity;

Or the resources can be supplied to the systems at a fixed concentration and rate (as in a chemostat),

\[ \Psi_{j} (R_{j}) = d(S_{j} - R_{j}) \,, \]

where \(d\) represents the flux of resources into and out of the system;

And/or the resources can be pulsed intermittently (e.g., in serial transfer—see examples below).

In the case of multiple resources, each resource is either treated as essential to consumer growth following Leibig’s law of the minimum, in which case,

\[ \mu_{i}(R_{1}, R_{2},...,R_{n}) = \min(\mu_{i}(R_{1}), \mu_{i}(R_{2}),..., \mu_{i}(R_{n})) \,; \]

or substitutable such that:

\[ \mu_{i}(R_{1}, R_{2},...,R_{n}) = \mu_{i}(R_{1}) + \mu_{i}(R_{2}) + ... + \mu_{i}(R_{n}) \,. \]

# Using `rescomp`

The main user function in rescomp is `spec_rescomp()`, which facilitates the definition and parameterisation of a consumer-resource model and the specification of simulation parameters. Function arguments include (but are not limited to):

- Number of consumers/resources
- Consumer functional response (type I, II or III)
- Resource dynamic (chemostat, logistic and/or pulsed)
- Resource type (substitutable or essential)
- Continuous or intermittent mortality (e.g., serial transfer)
- Delayed consumer introduction times
- Time dependent growth and consumption parameters

See `?spec_rescomp` for all argument options.

Examples demonstrated in this vignette include:

- One consumer (type 2) and one logistically growing resource

- One consumer (type 2) and one continuously supplied resource (i.e. chemostat)

- Two consumers (type 2) and one logistically growing resource

- Two consumers (type 1) and two substitutable resources in a chemostat

- Two consumers (type 1) and two essential resources in a chemostat

- Two consumers (type 2) and one externally pulsed resource

  - Continuous mortality
  - Serial transfer in batch culture (intermittent mortality)

- Two consumers (type 2) with time dependent consumption parameters and one continuously supplied resource

- Three consumers (type 1) on three essential resources generating an intransitive loop (i.e. rock paper scissors dynamics).

Additional examples illustrating other options and functionality can be found in the vignette “Reproducing studies in resource competition with rescomp”, including delayed consumer introduction times and specifying a model with consumer resource efficiencies rather than resource quotas.

## Type 2 consumer on a logistically growing resource

    library(rescomp)

    pars <- spec_rescomp(
      spnum = 1,
      resnum = 1,
      funcresp = funcresp_monod(mumax = crmatrix(0.12), ks = crmatrix(1)),
      ressupply = ressupply_logistic(r = 3, k = 2),
      rinit = 2,
      totaltime = 500
    )

Visualise functional responses with `plot_funcresp()`. Faceted by resources and time dependent parameterisations (when relevant).

    plot_funcresp(pars, maxx = 2)

The model can then be simulated with `sim_rescomp()`. `sim_rescomp()` is a wrapper to `deSolve::ode()` that takes the output from `spec_rescomp()` as its main argument.

    m1 <- sim_rescomp(pars)

`plot_rescomp()` produces a plot of the output dynamics (both consumers and resources).

    plot_rescomp(m1)

Note that the total simulation time and initial state variables specified with `spec_rescomp()` can be overidden in `sim_recomp()` with the arguments `totaltime`, and `cinit`, and `rinit`, respectively. This will nevertheless print a message to the console to check it is intentional.

    m1 <- sim_rescomp(pars, totaltime = 200, cinit = 30000)
    #> ℹ Overwriting `totaltime` in `pars`.
    #> ℹ Overwriting `cinit` in `pars`.

    plot_rescomp(m1)

## Type 2 consumer on a continuously supplied resource

    pars <- spec_rescomp(
      spnum = 1,
      resnum = 1,
      funcresp = funcresp_monod(mumax = crmatrix(0.12), ks = crmatrix(1)),
      mort = 0.03,
      ressupply = ressupply_chemostat(dilution = 0.03, concentration = 2),
      rinit = 2,
      totaltime = 300
    )

    m1 <- sim_rescomp(pars)

    plot_rescomp(m1)

## Two type 2 consumers on a logistically growing resource

    pars <- spec_rescomp(
      spnum = 2,
      resnum = 1,
      funcresp = funcresp_monod(
        mumax = crmatrix(0.35, 0.05),
        ks = crmatrix(1, 0.015)
      ),
      mort = 0.03,
      ressupply = ressupply_logistic(r = 1, k = 0.2),
      rinit = 0.2,
      totaltime = 2000
    )

    plot_funcresp(pars, maxx = 0.2)

    m1 <- sim_rescomp(pars)

    plot_rescomp(m1)

Owing to the trade-off in functional responses (gleaner-opportunist) and resource fluctuations generated by N2, the two consumers are able to coexist.

## Two type 1 consumers and two substitutable resources in a chemostat

    pars <- spec_rescomp(
      spnum = 2,
      resnum = 2,
      funcresp = funcresp_type1(crmatrix(
        0.09, 0.04,
        0.05, 0.08
      )),
      ressupply = ressupply_chemostat(dilution = 0.03, concentration = 1),
      mort = 0.03,
      essential = FALSE,
      totaltime = 1000
    )

    plot_funcresp(pars, maxx = 1)

    m1 <- sim_rescomp(pars)

    plot_rescomp(m1)

Resource partitioning, where each consumer is a better competitor (lower R*) for a different resource, results in coexistence.

## Two type 1 consumers and two essential resources in a chemostat

Using `pars$essential = TRUE` switches the previous parameteristion to essential resources while keeping all else equal.

    pars$essential <- TRUE
    m1 <- sim_rescomp(pars)

    plot_rescomp(m1)

There is no coexistence now because each species needs to have a larger impact on the resource that is most limiting its growth. For ‘consumer 1’ it is ‘resource b’, whereas for ‘consumer 2’ it is ‘resource a’. We can adjust consumer resource impacts via the resource quota matrix.

    pars <- spec_rescomp(
      spnum = 2,
      resnum = 2,
      funcresp = funcresp_type1(crmatrix(
        0.09, 0.04,
        0.05, 0.08
      )),
      quota = crmatrix(
        0.001, 0.005,
        0.005, 0.001
      ),
      ressupply = ressupply_chemostat(dilution = 0.03, concentration = 1),
      mort = 0.03,
      essential = TRUE
    )

    m1 <- sim_rescomp(pars)

    plot_rescomp(m1)

## Two type 2 consumers and one externally pulsed resource (continuous mortality)

External resource pulsing is handled using using the events system. The size of the resource pulse is specified as an argument to `event_res_add()`, while the frequency of the resource pulses is specified using `event_schedule_periodic()` and its `period` argument. To prevent any other mode of resource supply, `ressupply` is set to a constant 0 resource supply.

    pars <- spec_rescomp(
      spnum = 2,
      resnum = 1,
      funcresp = funcresp_monod(
        mumax = crmatrix(0.7, 0.05),
        ks = crmatrix(2, 0.015)
      ),
      ressupply = ressupply_constant(0),
      events = list(
        event_schedule_periodic(
          event_res_add(0.3),
          period = 100
        )
      )
    )

    plot_funcresp(pars, maxx = 0.3)

    m1 <- sim_rescomp(pars)

    plot_rescomp(m1)

## Two type 2 consumers and one externally pulsed resource (serial transfer with pulsed mortality)

Serial transfer in batch culture differs from the above scenario in that resource pulses are typically accompanied by large instantaneous changes in population density. For example, in a serial transfer experiment with bacteria, an experimenter might sample a fraction (e.g. 10 %) of the community every 24 hours and inoculate that fraction into fresh medium. As such, the consumer population density immediately following a transfer event will be equal to: \((1-M)N_{i}\), where \(M\) is instantaneous mortality fraction.

Similarly, the resources/medium is fractionally sampled, such that immediately after a transfer event the resource concentration will be equal to: \((1-M)R_{j} + MS_{j}\), where \(R\) is the resource concentration prior to transfer, and \(S\) is the resource concentration in the fresh medium.

The `event_batch_transfer()` function handles changes in both consumer and resource concentrations from a batch transfer event: `dilution` is the transfer fraction, \(1-M\), and `resources` is the concentration of resources in the fresh medium, \(S\). As for the previous example, we set the resource supply to zero, such that all resource input comes from the batch transfer events. In this case, we also set `mort` to zero, so that all consumer mortality is also provided by the batch transfer events.

    pars <- spec_rescomp(
      spnum = 2,
      resnum = 1,
      funcresp = funcresp_monod(
        mumax = crmatrix(0.2, 0.2),
        ks = crmatrix(0.3, 0.2)
      ),
      ressupply = ressupply_constant(0),
      mort = 0,
      events = list(
        event_schedule_periodic(
          event_batch_transfer(dilution = 0.2, resources = 1),
          period = 100
        )
      ),
      rinit = 0.6
    )

    plot_funcresp(pars, maxx = 1)

    m1 <- sim_rescomp(pars)

    plot_rescomp(m1)

Note, to reproduce the dynamics of a single batch culture experiment (i.e. without serial transfer) as potentially measured via optical density in a plate reader (negligible or unobserved mortality), simply set both `resspeed` and `mort` to zero.

    pars <- spec_rescomp(ressupply = ressupply_constant(0), mort = 0, totaltime = 500)
    m1 <- sim_rescomp(pars)

    plot_rescomp(m1)

## Two type 2 consumers with time dependent consumption parameters and one continuously supplied resource

Using time dependent consumption parameters is appropriate for capturing basic consumer-environment interactions (e.g. differential responses to temperature, pH, antibiotics etc.). Time varying parameters can be implemented with forcing functions. rescomp provides a convenient means of implementing forcing functions.

To do this, firstly, one or more time-dependent parameter must be defined, given in a list to the `params` argument of `spec_rescomp()`. The way in which the parameter changes over time may be defined using a variety of helper functions, such as `rescomp_param_sine()`, `rescomp_param_triangle()`, or `rescomp_param_square()` (different types of periodic variation). Secondly, some parameters of the model, such as the \(\mu_{\text{max}}\) and \(K_s\) of the functional response, must be made depedent upon the time-dependent parameters: this will most commonly be done using the `rescomp_coefs_lerp()` function, which linerarly interpolates the parameter values depending on the value of the time-dependent parameter.

    pars <- spec_rescomp(
      spnum = 2,
      resnum = 1,
      funcresp = funcresp_monod(
        mumax = rescomp_coefs_lerp(
          crmatrix(0.2, 0.5),
          crmatrix(0.3, 0.1),
          "env_state"
        ),
        ks = crmatrix(0.1)
      ),
      params = list(
        env_state = rescomp_param_square(period = 100)
      ),
      ressupply = ressupply_chemostat(dilution = 0.03, concentration = 0.1),
      rinit = 0.1,
      mort = 0.03,
      totaltime = 2000
    )

    plot_funcresp(pars, maxx = max(pars$rinit))

    m1 <- sim_rescomp(pars)

    plot_rescomp(m1)

The two consumers coexist due species-specific environmental responses leading to a temporal storage effect.

## Three type 1 consumers on three essential resources generating an intransitive loop (i.e. rock paper scissors dynamics).

    pars <- spec_rescomp(
      spnum = 3,
      resnum = 3,
      funcresp = funcresp_type1(crmatrix(
        0.11, 0.07, 0.045,
        0.05, 0.11, 0.07,
        0.07, 0.047, 0.1
      )),
      quota = crmatrix(
        0.2, 0.7, 0.4,
        0.4, 0.2, 0.7,
        0.7, 0.4, 0.2
      ),
      ressupply = ressupply_logistic(r = 10, k = 0.5),
      rinit = 0.5,
      mort = 0.015,
      essential = TRUE,
      totaltime = 2000
    )

    plot_funcresp(pars, maxx = max(pars$rinit))

    m1 <- sim_rescomp(pars)

    plot_rescomp(m1)