flowchart LR Susceptible --> Infected --> Recovered
Overarching questions in Consumer–resource dynamics
Consider a system with two species (\(R_1\) and \(R_2\)) that compete with one another for resources, but that are also predated on by the same predator (\(C\)).
Write a model that captures the interactions in this system.
For competing species \(R_1\) and \(R_2\):
\[\frac{dR_1}{dt} = r_1R_1(1-\alpha_{11}R_1 - \alpha_{12}R_2)\] \[\frac{dR_2}{dt} = r_2R_2(1-\alpha_{22}R_2 - \alpha_{21}R_1)\]
Modify these to account for consumer effects:
\[\frac{dR_1}{dt} = r_1R_1(1-\alpha_{11}R_1 - \alpha_{12}R_2)-a_1R_1C\] \[\frac{dR_2}{dt} = r_2R_2(1-\alpha_{22}R_2 - \alpha_{21}R_1)-a_2R_2C\]
The consumer equation also needs to reflect both sources of food:
\[\frac{dC}{dt} = e_1 a_1 R_1 C + e_2 a_2 R_2 C - mC\]
Figure from Bell and Fortier-Dubois 2017, Proc B
Such simplification enables predictions, e.g. what should we expect to happen if one of the herbivores is eliminated?
Of course, there’s some limit to how complicated a system we can reasonably model
One producer, an herbivore, and a carnivore
e.g. Kelp, sea urchins, and otters
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Assumptions:
Let’s start with some history
In 1926-27, Volterra and Lotka developed the model of consumer–resource dynamics
Assumptions
flowchart LR Susceptible --> Infected --> Recovered
Assumptions
To model this system we need three equations: \(\frac{dS}{dt}\), \(\frac{dI}{dt}\), \(\frac{dR}{dt}\)
\[\frac{dS}{dt} = \beta SI\]
\[\frac{dI}{dt} = \beta SI - \gamma I\]
\[\frac{dR}{dt} = \gamma I\]
\[\frac{dS}{dt} = \beta SI\]
\[\frac{dI}{dt} = \beta SI - \gamma I\]
\[\frac{dR}{dt} = \gamma I\]
flowchart LR Susceptible --> Infected --> Recovered Recovered --> Susceptible
flowchart LR Susceptible --> Infected --> Recovered Susceptible --> Vaccinated
flowchart LR Susceptible --> Infected --> Recovered Infected --> Dead
Some disease move through populations slowly enough that vital rates (birth and daeth rates) become important
Some diseases exist in which there are individuals who are carriers but not infected. These carriers don’t get sick but can get others sick.
Some diseases have an ‘Incubation’ period after infection (i.e. infected individual doesn’t immediately show signs of infection)
Stage/age structure: some diseases cause more effects in older or younger patients – combining matrix model approaches with SIR approach