Second try… lets add some density dependence

First off let me show off an updated version of the model I showed last time using the altered time step suggestion from the comments:

lvmod2That looks much nicer from a visual perspective (although I understand that it isn’t actually giving any different information).

Well my next step is to include density dependence in the prey species, adding a bit more realism to the model. I think I will take this opportunity to mention that I have been convinced by the ratio-dependence argument when it comes to the functional response in predator prey models. It seems clear to me from the evidence presented (see Lev Ginzburg and Roger Arditi’s new book How Species Interact, reviewed recently in Science here, and a different review here) that ratio-dependence is the way to go. I will post more on my thoughts on the predator prey models later though.

library(deSolve)
parameters2<-c(r=2,alpha=.1,e=.1,q=.5,K=600)
state<-c(N=50,P=20)
times<-seq(0,100,by=.1)
lvmodel2<-function(t,state,parameters2){ #with density dependence
with(as.list(c(state,parameters2)),{
#rate of change
dN<-(r*N*(1-N/K))-(alpha*N*P)
dP<-(e*alpha*N*P)-(q*P)

#return rate of change
list(c(dN,dP))
})
}

out2<-ode(y=state,times=times,func=lvmodel2,parms=parameters2)

plot(out2[,2],out2[,3],typ="o",pch=20,cex=.5,)
abline(v=.5/.01,col="blue") #dP/dt = 0
abline(h=2*((1/.1)-50/(600*.1)),col="blue") #dN/dt = 0

Which when run gives the output:

lvmod_dd

Good looking limit cycles converging to a stable equilibrium. I found it fairly interesting to mess with the initial parameter values to see how that changed the shape of the cycles. For example, increasing the carrying capacity 10x made this:

lvmod_dd2predictably increasing the equilibrium number of predators (although by less than I would have thought, and interestingly (although obviously when you examine the underlying equations) not altering the number of prey species. I think it is cool that changing the carrying capacity would make the population cycles take a longer time to reach equilibrium.

My next experiment with the ode solver was to incorporate the Arditi-Ginzburg functional response with prey density dependence. The code is here:

library(deSolve)
parameters3<-c(r=2,alpha=.5,e=.1,q=.25,K=600,h=.1)
state<-c(N=500,P=100)
lvmodel3<-function(t,state,parameters3){ #with density dependence and A-G fr
with(as.list(c(state,parameters3)),{
#rate of change
dN<-(r*N*(1-N/K))-((alpha*N)/(P+alpha*h*N))*P
dP<-(e*((alpha*N)/(P+alpha*h*N))*P)-(q*P)

#return rate of change
list(c(dN,dP))
})
}

times<-seq(0,100,by=.1)
out3<-ode(y=state,times=times,func=lvmodel3,parms=parameters3)

plot(out3[,2],out3[,3],typ="o",pch=20,cex=.5)

Playing around with the parameters generates a whole bunch of interesting results. I think I need to spend more time on coming up with realistic values for handling time, efficiency, and attack rate though. I am also not sure if I got the equations right (I cannot find the source I used for the equation) so I am going to look into that. I am a little curious to try out the various formulations of ratio-dependent functional responses that have been proposed and see how the outputs differ.

After that I want to figure out how to take what I have learned from modeling two species predator prey dynamics and see if I can expand that to three and more species (such as trophic chains or small food webs). I imagine that will be using population size vectors and matrices for the various parameters as well as slight reformulations of the equations to handle matrix inputs. But more on that later I suppose.

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9 Responses to Second try… lets add some density dependence

  1. Jeremy Fox says:

    Curious to hear why you think ratio dependence is the way to go. Strict ratio dependence is a *very* specific form of predator dependence, with pathological behavior at low predator densities (and low predator densities often occur in nature!). And if you say that models with ratio dependence can, in a phenomenological way, fit certain sorts of data, well, they’re hardly unique in that. Predator functional responses can depend on predator densities in all sorts of ways, for all sorts of reasons. Why not try to model those ways and reasons, instead of homing in on one very specific functional form with pathological properties?

    • Jon Borrelli says:

      You make a fair point. I will start with the fact that Lev Ginzburg is my current Ph. D. advisor, so I am constantly discussing with him his thoughts and views on prey vs. ratio dependence. I think the main argument for me, however, is when you examine the assumptions and the predictions of the two classes of models the standard Lotka-Volterra model tends to fall apart when applied to the data. This is not to say that the prey dependent view is completely useless, and Arditi and Ginzburg agree that there are cases where prey dependence is a better choice. The point they try to make, however, is that a majority of the time ratio dependence offers the better explanation. I think it is also important to note that they do not believe their model represents what could be called the truth either. The argument that Arditi and Ginzburg make in their papers and their new book is just that on the continuum of function responses, where g(N) and g(N/P) represent two extreme viewpoints the reality (what they say is g(N,P)) lies a bit closer to g(N/P).

      • Jeremy Fox says:

        One of the problems here, I think, is that you’re considering an overly-narrow range of alternatives to strict ratio dependence. See for instance the work of Mathew Leibold and others on how in simple food web models in which all predators have prey-dependent functional responses, turnover in species composition along productivity gradients leads to increases in total biomass on both the predator and prey trophic levels. And further, there’s actually good experimental evidence that that mechanism is actually what’s going on in many real-world systems.

        This is just one example, but I think it’s long been an issue in work on ratio dependence. Advocates of ratio dependence seem to want the predator functional response to do all the explanatory “work”. That seems to me to be focusing too narrowly, causing explanations having nothing to do with functional response shape to be ignored. It also seems to me to be a bad way of inferring what functional responses look like. If you want to know what functional responses look like, you should go out and measure functional responses. You shouldn’t try to fit alternative functional response models to data that can also be explained by various alternative mechanisms having nothing to do with functional response shape.

        I admit I have yet to read the new book, so perhaps these concerns are dealt with at length there…

      • Jon Borrelli says:

        Would you say then that the issue with focusing on the functional response is based on the assumption that the dynamics of predator and prey are fundamentally intrinsic to the system while there could be a number of external influences unrelated to the form of the functional response that impacts populations? I have not read Leibold’s work in this regard I will have to check it out.

      • Jeremy Fox says:

        Sorry, don’t know what you mean by “fundamentally intrinsic to the system” vs. “external influences”. I thought that, in a multispecies system, that all the species together comprised the dynamical system.

        Mathew Leibold’s stuff is just one bit of a much larger and broader body of work. There’s a whole world of dynamically-oriented food web ecology out there, from all sorts of people–Mathew, Peter Abrams, Bob Holt, Jon Chase, Jim Grover, and many, many more,. It’s concerned, yes, with the dynamical effects of functional response shape, but also with all sorts of other factors that affect every aspect of the behavior of systems in which some species eat other species.

        Just my two cents, but I think it’s worth keeping separate two sorts of questions that seem to me to have been conflated by folks arguing for ratio dependence, at least in the past. One is “what do predator functional responses look like?” The best way to answer that question, it seems to me, is via direct empirical evidence: people ought to do the experiments needed to measure functional response shape. And indeed, Roger, Lev, and Peter Abrams have all quite rightly called for collection of this sort of data. The other sort of question is “What’s the explanation for features X, Y, and Z of data on the dynamics and abundance of species and entire trophic levels?” It seems to me that “it has to do with the shape of the functional response” is only one among many possible answers to that second sort of question. And so if you’re truly serious about answering that sort of question (as opposed to merely trying to accumulate very indirect and therefore very weak evidence about functional response shapes), you need to consider the full range of possible answers.

        Put another way, I think there’s a tension between the various justifications that have been offered for ratio dependent models over the years. On the one hand, they’re sometimes justified via mechanistic arguments about what determines predator functional response shape. On the other hand, they’re sometimes justified as a flexible, phenomenological way to implicitly account for the dynamical effects of all sorts of unspecified underlying mechanisms, including mechanisms having little or nothing to do with functional response shape per se. I think those two justifications are in tension because they suggest very different lines of research. Insofar as you think ratio dependent models are mechanistically justified, then you’re going to want to go out (or have others go out) and conduct the detailed mechanistic observations and experiments needed to test your underlying mechanistic claims. And you’re not going to care at all about whether ratio dependent models can or cannot be fit to, or estimated from, population dynamic data or data on the total biomasses of different trophic levels or whatever. Because, as argued above, that sort of curve fitting is at best really weak indirect evidence for the actual mechanisms that govern functional response shape. Conversely, if you think of ratio dependent models as just a convenient, purely phenomenological summary of any and all unspecified underlying mechanisms affecting predator-prey dynamics, then you’re going to care a lot about their ability to fit predator-prey dynamics, and not at all about the results of direct, detailed mechanistic measurements of predator functional response shapes. Perhaps you and Roger and Lev might disagree with me here, as I say I haven’t yet been able to free up time to read their new book. But I don’t think this is a case where one can have one’s cake and eat it too.

      • Jon Borrelli says:

        So, for example, Arditi and Ginzburg posit that the underlying mechanism for ratio-dependence is interference competition between predators for prey. If I understand your argument, it is not enough that the data on dynamics fits the model with ratio dependent functional response better than the model with prey-dependence. Instead we should attempt to find evidence to justify the mechanism. So it would seem that fitting the model to data would be necessary, but not sufficient evidence to convincingly demonstrate that ratio-dependence is an important factor in the dynamics of predator and prey. Instead of fitting the model to data we should go out (“or have others go out”, I like that bit haha hooray theoretical ecology) and measure to what extent are predators interfering with each other. I do believe that Arditi and Ginzburg make some predictions to that effect, and may have some empirical stories to back them up as well (I also need to read the book again to jog my memory). I think, for example, they start off comparing prey and ratio dependence to Monod and Contois two different equations for bacterial growth which is a directly parallel example.

        I also like to think that at the very least that the ratio dependent view provides an interesting framework within which we can gain greater understanding of trophic dynamics. For example, in the realm of the phenomenological, adding ratio dependent functional responses to food web models increases their fit to real data (although granted you can do this with other modifications to the models such as foraging adaptation and body size). Nonetheless I think that within the umbrella of factors that affect dynamics of predators and their prey, the functional response should be considered. I think that your point of examining the full range of factors is also important.

      • Jeremy Fox says:

        p.s. Let me say that I really appreciate you taking the time to discuss this stuff, Jon. I do have strong views on ratio dependence, but I hope the frankness of my comments doesn’t come across as an attempt to browbeat you or as indicating that my mind is closed to anything you or Roger or Lev might have to say. All I can do as a starting point is lay out my current views; doesn’t mean I’m not open to changing them. And I do mean it when I say I’m trying to find time to read the new book. I like being challenged and pushed to think hard and it sounds like the book will do that for me.

      • Jon Borrelli says:

        It is my pleasure, I have quite enjoyed this discussion!

  2. Jon Borrelli says:

    I would also say that to your point of: “Why not try to model those ways and reasons, instead of homing in on one very specific functional form with pathological properties?” I agree and I think that in my future research I would like to explore a range of potential functional responses that span the spectrum. By not limiting ourselves to one form or the other I think that we can better understand the dynamics between predator and prey.

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