Refuge impact on growth rate

Question

I have a question related to the refuge effect in ecological models. The assumption is that the population of hosts/prey is divided into two groups. One group consists of individuals that are safe from parasitoids/predators and another group consists of individuals that are exposed to parasitoids/predators and can get parasitized. Does it affect the growth rate in the sense that hosts that are in refuge have a bigger growth rate than those that are not in refuge? Are there some known experimental results, some papers or books that deal with this topic?

Answer 1

The answer to this depends a lot on what kind of question you're asking. Adding more context/clarifying your question might help you get more useful answers.

If you run a short-term experiment starting with the same population sizes in each region, and little mixing between populations, the protected population must grow faster (given that the protected region is the same in every way as the exposed region except for being protected from natural enemies).

Of course, if you don't have enough statistical power (there isn't very much predation even in the exposed region, or your population counts are not very accurate, or there is lots of other random variation in the growth rate) then you might not be able to detect this difference.

There are lots of potential confounding effects:

• how smart are the prey? Do they preferentially move into the refuge (in which case the protective effect of the refuge might be overestimated)?
• ... or do they prefer places with lower conspecific densities (in which case they might preferentially move out of the refuge as predators remove conspecifics in the exposed region; you might underestimate the protective effect of the refuge)
• or do conspecific densities affect survival or fecundity in other ways? (If the protected individuals die less from predation, this might be compensated by other effects of crowding.)
• how do the natural-enemy (predator/parasite/parasitoid) densities respond to changes in host density?
• are the refuge and the protected area really identical in all other ways? If this is an observational study, maybe the prey species is already making a tradeoff between areas on the basis of predation risk balanced with some other benefit (such as greater resource availability)

There is a lot of research about all of these complexities (you might start by doing a literature search on "ideal free distribution" and "predator", or "landscape of fear"; however, most of this research is about short-term redistribution rather than long-term growth rates)

If none of these complications is occurring (in particular, no other environmental differences and no differences due to host densities in each area) then we could also say that the fitness (i.e., expected number of lifetime offspring) of hosts in the protected area would also be expected to be higher than that of hosts in the exposed area (this might be what you mean by "growth rate").

You might want to dig into the literature on marine protected areas, where researchers consider how population dynamics work inside and outside spatial regions where fishing is excluded (which is similar to protection from predators or parasites).

From a very narrow mathematical biology perspective, if the two populations are coupled (i.e. individuals can move back and forth between refuge and non-refuge areas), then there aren't separate growth rates for the two subpopulations — in the long run, both will grow at the same (exponential) rate as the whole population (discounting density-dependent effects). Consider the simplest possible model:

$$ \begin{split} N_e' & = (r-p-m)N_e + m N_p \\ N_p' & = (r-m) N_p + m N_e, \end{split} $$ where $N_e$ and $N_p$ are the "exposed" and "protected" population sizes (and $N_e'$, $N_p'$ are their derivatives/instantaneous growth rates); $r$ is the intrinsic growth rate (in the absence of predators); $p$ is the per capita predation rate; and $m$ is the migration rate, the per capita rate at which individuals move back and forth between protected and exposed areas. (This model is horribly unrealistic because it assumes that all of the per capita rates are constant — neither the intrinsic growth rate of the prey, nor the attack rate of the predators, nor the behaviour of the prey, can change as the population densities in both regions change.)

We can do the typical mathematical analysis of this system:

• find the Jacobian of the system

$$ J = \left( \begin{array}{cc} r-p-m & m \\ m & r-m \end{array} \right) $$

• calculate its eigenvalues, $r-m - \frac{p}{2} \pm \frac{\sqrt{4 m^2 + p^2}}{2}$
• conclude that the overall population size (and both components) grow, asymptotically, at an exponential rate given by the largest eigenvalue (the positive root above), unless $m=0$ (in which case the protected population grows at rate $r$ and the exposed population grows at rate $r-p$).
• if you want to know the relative contribution of the two subpopulation densities to the whole population (attained asymptotically), calculate the eigenvector associated with the dominant eigenvalue and scale its magnitude to 1.0.