# How is the carrying capacity of a logistic growth model calculated?

I am reading a book in epidemiology where the carrying capacity for a standard logistic growth rate is given by

K = (b - delta) / gamma


where:

K is the carrying capacity
b is the birth rate
delta is the death rate


but what is gamma?

Thank you

• Your equation is not a difference equation (discrete time) nor a differential equation (continuous time). It therefore cannot describe a population growth. I don't know what the authors may have intended with this equation but we would need more information to address that question (book reference, page number, full description of the equation, etc...) May 2, 2020 at 8:19

@Remi.b is correct that you haven't given us very much information, but I think we can reconstruct what's going on. Suppose the population growth rate is written out as

$$\frac{dN}{dt} = N ( b - \delta - \gamma N)$$

then the equilibrium (carrying capacity) occurs when $$N>0$$ and $$dN/dt=0$$, i.e. $$b - \delta - \gamma K = 0$$. Solving this for $$K$$ gives $$(b-\delta)/\gamma$$ (as you stated).

So what is $$\gamma$$? It is the decrease in the per capita growth rate per unit of increase in population density, or more biologically speaking it's the decrease in the birth rate or the increase in the death rate per unit of increase in population density. This could be due to increased competition for resources, or decreased environmental quality, or attraction of predators, or ...

Another, vaguer way to say this would be to call it the "effect of density-dependence".

Given the equations you list in the comments, with a $$-\gamma N$$ term included in the per capita growth rate for each compartment ($$S$$, $$E$$, $$I$$), we can say more specifically that $$\gamma$$ determines the rate of density-dependent increase in the ("natural" or disease-independent) death rate per unit population density. This parameter would not typically be broken down into lower-level processes.

• sorry, I thought it was a standard measure, thus no information needed -- and in fact, you reverse-engineered. K was mentioned in a SEIRD model. Is gamma a constant or is it derived itself from more basic data? Thank you May 3, 2020 at 12:49
• Can't really answer that question without more context. Can you give a link to the source, or post a screenshot of the relevant part of the text, as part of your question? May 3, 2020 at 20:51
• the context is a SEIR model with: May 4, 2020 at 21:26
• the context is a SEIR model with: dS/dt = (b-delta-gamma * N)*S - beta * S * I; dE/dt = beta * S * I - (sigma + delta + gamma * N) * E; dI/dt = sigma * E - (delta + deltaI + gamma * N) * I; with b = birth rate, delta = death rate, 1/sigma = latency, gamma as you said, N = total population. May 4, 2020 at 21:48
• I'm curious: can you say what book you're using? May 4, 2020 at 22:52