A simple model of consumer-resource dynamics

Building a simple consumer–resource system

• Consider the dynamics of a “consumer” species \(C\) and a “resource” species \(R\)

• The resource species has plenty of resources available and can grow exponentially…

• … but it is suppressed by the consumer.

• The consumer is a specialist (only eats the “resource” species)

• (If there is no resource around, the consumer population dies out)

• The consumer loses energy at some rate (maintenance cost)

• How should the abundances of consumers and resources change over time? (\(\frac{dR}{dt}\) and \(\frac{dC}{dt}\))

Resource species dynamics

• The resource species has plenty of resources available and can grow exponentially…
• … but it is suppressed by the consumer.

\[\frac{dR}{dt} = \text{exponential growth - loss to consumer}\]

\[\frac{dR}{dt} = rR - aRC\]

• \(R\) is the abundance of the resource species
• \(r\) is the intrinsic growth rate of the resource
• \(a\) is the attack rate of the consumer on the resource
• \(C\) is the abundance of the consumer species

Consumer species dynamics

• The consumer is a specialist (only eats the “resource” species)
• The consumer loses energy at some rate (maintenance cost)

\[\frac{dC}{dt} = \text{growth from consumption - energy loss}\]

\[\frac{dC}{dt} = eaRC - mC\]

• \(C\) is the abundance of the consumer species
• \(a\) is the attack rate of the consumer on the resource
• \(e\) is the efficiency with which consumer uses the energy it gets from resource
• \(R\) is the abundance of the resource species
• \(m\) is the maintenance cost of the consumer species

Lotka-Volterra consumer-resource system

\[\frac{dR}{dt} = rR - aRC\]

\[\frac{dC}{dt} = eaRC - mC\]

Some “obvious” takeaways:

• If consumer species is absent (\(C = 0\)), the resource species grows exponentially
• If the resource species is absent (\(R = 0\)), the consumer species dies out

Lotka-Volterra consumer-resource system

\[\frac{dR}{dt} = rR - aRC\]

\[\frac{dC}{dt} = eaRC - mC\]

Some more takeaways:

• If the resource has high intrinsic growth rate (\(\uparrow r\)), it is more likely to grow even when many consumers are present
• If the consumer attacks the resource at a very high rate (\(\uparrow a\)), resource populations will be suppressed, and consumer population grows more rapidly
• If the consumer is very efficient at getting energy from the resource (\(\uparrow e\)), the consumer populations grow more rapidly
• If consumer has high maintenance cost (\(\uparrow m\)), consumer populations will be suppressed

Equilibrium in C-R systems

Given the consumer–resource dynamics equations:

\[\frac{dR}{dt} = rR - aRC\]

\[\frac{dC}{dt} = eaRC - mC\]

Where does equilibrium occur?

• How can we illustrate this in state-space?

Equilibrium analysis of Resource

\[\frac{dR}{dt} = rR - aRC = 0\]

\[rR = aRC\]

\[C^* = \frac{r}{a}\]

Interpretation: There is a set abundances of the consumer \(C\) that would cause the resource \(R\) to be at equilibrium.

• Below this threshold, the resource population can grow (there are enough new births in the resource to overcome what is lost to the consumer)

• Above this threshold, the resource population shrinks (there are not enough new births in the resource to overcome what is lost to the consumer)

Equlibrium analysis of consumer

\[\frac{dC}{dt} = eaRC - mC = 0\]

\[eaRC = mC\]

\[R = \frac{m}{ea}\]

Interpration: There is a set abundance of the resource \(R\) that would cause the consumer to reach equilibrium

• Below this threshold, the consumer population shrinks
• Above this threshold, the consumer population can grow

Putting the isoclines together

Extending beyond 1 resource and 1 consumer

(note: you won’t be asked to know the following models for class activities)

Model eqns:

\[\frac{dR}{dt} = rR - aRC\]

\[\frac{dC}{dt} = eaRC - mC\]

Model eqns:

\[\frac{dR_1}{dt} = r_1R_1 - a_{11}R_1C_1\]

\[\frac{dC_1}{dt} = e_1a_1R_1C_1 - m_1C_1\]

\[\frac{dR_2}{dt} = r_2R_2 - a_{22}R_2C_2\]

\[\frac{dC_2}{dt} = e_2a_2R_2C_2 - m_2C_2\]

Model eqns?

(Logistic growth in resource)

Model eqns?

(Competition between resource species)

Model eqns?

Figure from Bell and Fortier-Dubois 2017, Proc B