1. ANALYSIS OF MULTI-SPECIES ECOLOGICAL AND EVOLUTIONARY DYNAMICS
Ecole Normale Supérieure, Paris
December 9-13, 2013 2.
Simple models of competition and mutualism (F. Dercole)
The Lotka-Volterra competition model. Symmetric vs asymmetric competition. Equilibria and isoclines. The principle of competitive exclusion. Transcritical bifurcations. A simple model of mutualism. Obligate vs non-obligate mutualism. Equilibria and isoclines. Saddle-node bifurcation.
Further readings
Encyclopedia of Theoretical Ecology, Univ. California Press, 2012, pp
Proc. Roy. Soc. Lond. B (2002) 269:
OK, in this lecture I want to introduce and analyze with you two simple ecological models, the L-V competition model and a model of a mutualistic interaction

2. The Lotka-Volterra competition model
Competition within one population (the logistic model)
is the intrinsic (or initial) per-capita growth rate
is the per-capita competition mortality
is the carrying capacity
Competition within two populations
(adimensional) competition coefficients
- I am quite sure that you have already encountered the L-V competition model in other courses, but this lecture is meant as a warm-up and to introduce notation and a few basic concepts in systems’ theory that we will be using throughout the course
- Let me start with the simplest competition model, that is also the historical origin of modeling competition, namely logistic model
- We have a linearly density dependent term in the growth rate, which describes the balance between natality and natural mortality
- and we have a quadratic mortality describing the encounters of individuals in pairs and the fact that they compete, or fight for example, for the same resource
- Factoring the linear growth rate, we can easily see that r/c is the equilibrium density that can be sustained by the resource, and it is called the carrying capacity of the environment
- If we now consider the competition between two populations, that we think e.g. belonging to different species, we can write the L-V competition model, where each population grows logistically in the absence of the other and suffers an extra mortality if the other is present, and this mortality is proportional to the product of the densities that measures the encounter rate between individuals of the two populations
- Again factoring the linear growth rate, we can highlight the carrying capacities of the two species and define the adimensional competition coefficients, that are defined by the competition suffered due to the other population normalized by the competition within the population
- We say that competition is symmetric if … and asymmetric otherwise. Recall that alpha12 represents the mortality suffered by population 1 due to pop 2, so that alpha12 > alpha21 means that…
symmetric competition
asymmetric competition favoring population 2 / 1

3. Competition within two populations
Equilibria and isoclines
equilibria :
and
the curves in the state plane
where and
isoclines :
- Now, let’s make a little exercise for computing the equilibria of the model. So we need to find the solutions (n1,n2) of the two stationary conditions n1dot=0, n2dot=0. For 2-dim systems, as the one we are considering, it is useful do it graphically using the so-called isoclines.
- The isoclines are lines along which… Here I drawed in red / blue … that for the L-V competition model they are simply straight lines. The equilibria are at the intersections between red and blue curves, so we have 4 equilibria…
- The isoclines are also useful to deduce the direction of trajectories. Instead of looking where n1dot and n2dot are zero, we check on which side of each isocline n1dot and n2dot are positive or negative. Here, for example, above both isoclines both derivatives are negative, so that trajectories go left and down…
- Recall that (n1dot, n2dot) is the vector tangent to trajectories, so trajectories intersect the blue isocline vertically, because n1dot=0, and the red isocline horizontally
- and note that this analysis often tells us the stability of equilibria, as in this example for example, where we see that the equilibrium of coexistence is a saddle, attracting along… and repelling…, while…
- Thus, here initial conditions matter, because depending on… stable manifold of the saddle
- This scenario well describes the principle of competitive exclusion, that basically states that competition makes impossible the coexistence on the same resource
- But is this the only possible scenario for the L-V competition model? Actually not. We see again from the isoclines that if we change the parameters these two points on the n1 axis, as well as the other two on the n2 axis can exchange, so we see three other possible scenarios, by switching this pair, this pair, or both
- Let’s now play with K1. This is rather easy because K1 moves up and down only the n1dot-isocline. So increasing K1, the blue equilibrium moves up and collide with the green one at (0,K2), and if we increase K1 even further the coexistence equilibrium goes outside the positive quadrant (cambiare lucido)
the direction of trajectories :
the principle of competitive exclusion
(Hardin G., Science 131, 1960; Gause G.F., Williams&Wilkins, 1934)

4. Transcritical bifurcations (see f.r. 1)
geometric view
: collision of two equilibria, as a parameter is varied, which “exchange stability”
algebraic view
: a zero eigenvalue in the system’s Jacobian
- and this is the situation we get (indicare quella centrale)
- The transition is a so-called transcritical bifurcation. Geometrically it involve the collision of two equilibria, that pass one through the other and exchange stability. But bifurcations can also be characterized algebraically, and actually the algebraic conditions are those used in the software for the numerical analysis. For the transcritical, the condition is the presence of a zero eigenvalue… here, for example, by increasing K1, the positive eigenvalue of the saddle goes to zero, as well as one of the negative eigenvalues of this stable equilibrium. At the bifurcation the equilibria coincide and both have a zero eigenvalue. After the bifurcation the equilibrium on the n2-axis is unstable, because its eigenvalue went from negative to positive, while the coexsistence equilibrium, that is not anymore visible here, is stable…
- But we can also forget about eigenvalues and just look at the direction of the trajectories given by the study of the isoclines… (K1,0) is still stable but much further on the right in this figure
- Similarly, if we decrease K1 starting back from the first panel, the isocline goes down and we have the collision with the equilibrium (K1,0), which becomes unstable and all trajectories after the bifurcation go to (0,K2)
- So, if we make a 3d-plot of this exercise, we draw the picture that has been anticipated at the end of the first lecture
- Note also that we have seen two more competition scenarios, the dominance of species 1, and the dominance of species 2

5. Four possible scenarios (state portraits)
coexistence
dominance-2
dominance-1
mutual exclusion
- The fourth scenario is the coexsistence at a stable equilibrium, that you get from the mutual exclusion by reducing the competition…
- Here you see the typical trajectories, note that they intersect the isoclines vertically and horizontally
- and here summarized you have the conditions under which the four scenarios occur

6. Back to the principle of competitive exclusion, consider the case of
symmetric competition with
Mutual exclusion is the resulting scenario when competition is sufficiently strong
- Before switching to mutualism, I want to leave you another exercise on transcritical bifurcations that better shows the principle of competitive exclusion. We consider symmetric competition, with K1 > K2, and increase competition. Note that we have coexistence at low competition and mutual exclusion if competition is sufficiently high

7. A simple model of mutualism
Two species, e.g. flowers and pollinating insects, with densities and
The per-capita rates of commodities trading are inheritable phenotypes and
thus is the prob. that an individual of species 2 receives a benefit from species 1 in the time interval
similarly for
There is intra-specific competition for commodities, as well as for other resources
The mutualism is obligate
A simple model (see f.r. 2)
- We now switch to mutualism, and I like to present you a simple model developed by Regis Ferriere, Sergio Rinaldi and others, on which I also worked during my Ph.D. to catalog all bifurcations
- There are two mutualistic species, the typical example is flowers and their pollinating insects and we assume that x1, x2 are two inheritable phenotypes that measure in terms of energy the per-capita rates of commodities trading between the two species
- Per-capita means ‘per-individual’, we should better say per-unit of density, but it is typical to say per-capita
- x1 and x2 are here constant parameters but in the lab tomorrow afternoon we will build an evolutionary model for the evolution of these two traits
- we also assume intra-specific competition for the commodities exchanged by the two species, as well as for other resources
- And this is the model. Here the parentheses collect the per-capita growth rates of the two populations (since we have factored the density of the population), which are composed of three terms:
- r (1 and 2) are the balance between the natality that is possible without the mutualistic interaction and natural mortality, and the balance is assumed to be negative because the mutualism is obligate …
- d1 and d2 describe the competition for other resources, namely different from the commodities coming from the partner species … and this competition is intra-specific
- and finally the third terms are the extra natality thanks to the mutualistic interaction: x2n2n1 is the total help, in terms of energy for reproduction, received by species 1, that is discounted by the factor (1-c1n1) because individuals compete for the mutualistic help. At the point that when the density is too high the induced mortality overcome the benefit of the interaction… and similarly for species 2
- r1 and r2 are assumed to be increasing (even more than linearly actually) to describe the increasing costs, in terms of reproduction, of being more altruistic. There is therefore a trade-off between fitness and being good mutualists
- and these are the constant parameter of the model that are all positive
where and are nonnegative increasing functions
and , , , , are positive constant parameters

8. Equilibria and isoclines
the direction of trajectories :
The saddle-node bifurcation (see f.r. 1)
The evolution set
geometric view
: collision and disappearance of two equilibria
algebraic view
: a zero eigenvalue in the system’s Jacobian
- Let’s again look for the equilibria by using the isoclines. The axes are again isoclines, due to the density factored in front of the per-capita growth rate (tornare al lucido precedente; this is typical of all population models), and then we have two branches of hyperbola
- so depending on the fact that the two branches intersect each other or not, we can have two positive equilibria or just the extinction at the origin
- If we now look at the direction of the trajectories we see that the origin is always stable, so that if the initial densities are too small the mutualistic benefit is not enough to sustain the two populations
- and, when present, the intermediate equilibrium is a saddle and its stable manifold (qui far apparire le traiettorie) separates the initial conditions allowing persistence from those leading to extinction
- The transition between the two cases is obviously a bifurcation, and this is called the saddle-node bifurcation, because a saddle and a node collide, while moving a parameter and disappear
- This bifurcation occurs if we reduce either x1 or x2. This is rather obvious, because the mutualism is obligate. From the model we see that e.g. if we reduce x1 then the red branch moves right, while if we reduce x2 then the blue branch moves up.
- Interestingly the bifurcation is reached also by increasing x1 and/or x2, due to the trade-off between altruism and reproduction or mortality. From the model we see that the isoclines move up and right by increasing x1 and x2…
- Once we are at the bifurcation, detected e.g. by increasing x1, we can continue the bifurcation condition by varying x1 and x2 simultaneously, we numerically produce the saddle-node bifurcation curve in two parameters. And this is the result, that you will produce tomorrow in the lab
- It is a closed curve that defines what we call the “evolution set”. Inside this set evolution can take place, because the two species can coexist at the positive stable equilibrium, so that on the evolutionary time-scale good genetic mutations replacing the former residents can accumulate and produce the evolutionary change of the traits x1 and x2 that we have assumed to be inheritable.
- So, inside the evolution set we will learn how to generate evolutionary trajectories for the adaptive traits x1 and x2