G-function without abiotic components

In this case, the equation system to be solved remains exactly as in equation (6.4.1).

We study the cases investigated in section 2.2.1: a case where there is only the public benefit to QS, one with private benefits B(v_i) and one with a strategy-dependent death rate.

G(v_i, v, b) = B(v, b)·C(v_i)−µkbk₁

We have seen in our calculations that v_i = 0 is the only stable strategy in this situation. We should therefore have vi → 0, and ifB_min = 0, bi →0. Yet, when looking at the simulation results for 3000 h as displayed in figure 6.6, we notice that this convergence is very slow indeed. A quick calculation of∂₁G and G for our parameter values confirms that both are close to zero. So even though QS is theoretically unstable, both cheaters and producers persist alongside each other for a very long time, albeit at different densities.

G(v_i, v, b) =

B(v, b) +B(v_i)

·C(v_i)−µkbk₁

In section 2.2.1 we have focussed on Hill-terms for B(vi) and realized that the long-term behaviour is critically dependent on the Hill coefficienth. Figures 6.7 to 6.9 show the results for h= 1,2 or 3.

As predicted, the behaviour in these cases is quite different. For h ≤ 2 we postulated that there can only be one positive stationary point for v and if it is stable, then the zero solution is unstable. This is the case for the parameters we have calculated. Additionally, we find that for h= 1 the zero solution is not a stationary point. Thus both strategies converge towards the stable positive equilibrium, one from above, the other from below. Population 2, which started out as cheaters, gains QS functionality, while there is reduced production from population 1. Since reducing production is slower to reach the stable point in this parameter constellation, population 1 succumbs to the population pressure from population 2 and dies out (see figure 6.7). This happens on a rather short time frame of less than 200 h.

Forh= 2, we can see from figure 6.8 that once again the strategies converge to a positive value. But this time the zero solution is a stationary point, although an unstable one. Hence population 2 cannot gain QS functionality by starting out

Figure 6.6: Long-term behaviour of two populations with different start strategies, using a G-function without additions. Both populations numbers are slowly converging towards zero, with WT having the lower CFU.

Figure 6.7: Evolution of two populations with different start strategies, using a G-function with private benefit and a hill factor of h = 1. Note the shorter time-scale in this plot. Population 1 dies out rather quickly, while population 2 gains the QS functionality, albeit at a low level, and remains at a stable population level.

Figure 6.8: Evolution of two populations with different start strategies, using a G-function with private benefit and a hill factor of h = 2. In this scenario, population 1 reduces its QS strategy to a lower, but stable value. Population 2 is unable to compete and dies out.

Figure 6.9: Evolution of two populations with different start strategies, using a G-function with private benefit and a hill factor of h= 3. Both population 1 and 2 are coexisting in a stable way with similar CFU. One population is QS active while the other is not.

withv₂ = 0. The end result is the extinction of population 2, while population 1 remains stable.

All in all, for h≤2 we find that one population is driven to extinction, while the other stays at a stable level with QS intact at lower levels.

When h >2, we know from section 2.2.1 that there might be a stable positive strategy in addition to a stable zero solution. We find this to be the case for h= 3 for our parameters, as one can see in figure 6.9. This means that both population 1 and population 2 remain at stable population and strategy levels, with population 1 practising QS while population 2 consists of cheaters. In this scenario, producers and non-producers can live side-by-side indefinitely.

Additionally, for h= 3, there might exist unstable positive stationary strategies, depending on parameter values. For simplicity’s sake, we show the effect for one population only and set B_min = 1, K = 0.5 as well as disable cost dependency.

As a result, when set to this unstable state both population and strategy remain constant, but tend towards either zero or the positive evolutionary stable strategy (ESS) if perturbed. If the perturbation is not spatially homogeneous, the resulting stationary state might not be homogeneous as well. One example for such an effect is shown in figure 6.10.

G(v_i, v, b) = B(v, b)·C(v_i)−µ(v_i)kbk₁

Instead of a direct private benefit we have also discussedG-functions with strategy-dependent death rates µ(v_i) in section 2.2.1. The explicit function examined in detail was

µ(v_i) = (µ_max−µ_min) e^−dvi2 +µ_min.

We know that the behaviour largely depends on the relation between K and d and between B(v, b),kbk₁, ∆µandK. In order to change the latter of the two relations, one would have to change some of the “main” parameters we have kept constant so far. In addition, it is difficult to calculate this value before running the simulation. It is thus easier to change the relation between K and d, even though we have raised some concerns about the caseK > d in section 2.2.1. Our primary concern we voiced was about the instability of the positive stationary point ¯v_i and the subsequent divergence of the strategy value for large v_i. But in praxis, this instability hardly matters if ¯v_i is large enough. In fact, one could even relax our initial assumption from

∃v^∗ : ∂₁G(v^∗, v, b)<0 ∀v_i ≥v^∗ to

∃v^∗ >maxⁿv_i⁰

∞

o: ∂₁G(v^∗, v, b)<0

Figure 6.10: Time evolution of a strategy starting out in an unstable stationary point with and without perturbation. Without perturbation, the strategy value remains in the unstable point. With the addition of a small perturbation, it converges towards the nearest ESS. If the perturbation is sine-shaped, as in this case, the resulting strategy limit is spatially inhomogeneous.

without implications. As such, we can safely consider a parameter set with K > d, as long as ¯v_i >max{kv_i⁰k_∞}.

Since we have set K = 2.459, the cases d = 2 and d = 3 are of particular interest. Additionally, we need to specify µ_max as well asµ_min. We expect that the constant death rate of µ= 7.24×10⁻⁷ is the maximal death rate for this model.

For the minimal death rate we roughly assume a reduction to one tenth, so that µ_min = 7.24×10⁻⁸.

Ifd = 3, we have K < d and a behaviour very similar to a private benefit with h = 2: There is a positive stable equilibrium point for v_i to which population 1 converges, while the strategy of population 2 is caught in the unstable zero point.

In accordance with these results, population 2 dies out while population 1 remains at a stable level. The results are shown in figure 6.11.

If d = 2, on the other hand, we have K > d. Since a short calculation for our parameter values yields ¯v_i = 108, the critical point is indeed far away from the interesting strategy value domain. As expected, the populations behave as in the case without additions, with both populations declining very slowly towards extinction (see figure 6.12).