
Citation: Patrick De Leenheer, Martin Schuster, Hal Smith. Strong cooperation or tragedy of the commons in the chemostat[J]. Mathematical Biosciences and Engineering, 2019, 16(1): 139-149. doi: 10.3934/mbe.2019007
[1] | José F. Fontanari . Cooperation in the face of crisis: effect of demographic noise in collective-risk social dilemmas. Mathematical Biosciences and Engineering, 2024, 21(11): 7480-7500. doi: 10.3934/mbe.2024329 |
[2] | Alain Rapaport, Jérôme Harmand . Biological control of the chemostat with nonmonotonic response and different removal rates. Mathematical Biosciences and Engineering, 2008, 5(3): 539-547. doi: 10.3934/mbe.2008.5.539 |
[3] | Cheng-Hsiung Hsu, Jian-Jhong Lin, Shi-Liang Wu . Existence and stability of traveling wavefronts for discrete three species competitive-cooperative systems. Mathematical Biosciences and Engineering, 2019, 16(5): 4151-4181. doi: 10.3934/mbe.2019207 |
[4] | Hal L. Smith, Horst R. Thieme . Chemostats and epidemics: Competition for nutrients/hosts. Mathematical Biosciences and Engineering, 2013, 10(5&6): 1635-1650. doi: 10.3934/mbe.2013.10.1635 |
[5] | Manel Dali Youcef, Alain Rapaport, Tewfik Sari . Study of performance criteria of serial configuration of two chemostats. Mathematical Biosciences and Engineering, 2020, 17(6): 6278-6309. doi: 10.3934/mbe.2020332 |
[6] | Jianquan Li, Zuren Feng, Juan Zhang, Jie Lou . A competition model of the chemostat with an external inhibitor. Mathematical Biosciences and Engineering, 2006, 3(1): 111-123. doi: 10.3934/mbe.2006.3.111 |
[7] | Nahla Abdellatif, Radhouane Fekih-Salem, Tewfik Sari . Competition for a single resource and coexistence of several species in the chemostat. Mathematical Biosciences and Engineering, 2016, 13(4): 631-652. doi: 10.3934/mbe.2016012 |
[8] | Ryusuke Kon . Dynamics of competitive systems with a single common limiting factor. Mathematical Biosciences and Engineering, 2015, 12(1): 71-81. doi: 10.3934/mbe.2015.12.71 |
[9] | Bo Lan, Lei Zhuang, Qin Zhou . An evolutionary game analysis of digital currency innovation and regulatory coordination. Mathematical Biosciences and Engineering, 2023, 20(5): 9018-9040. doi: 10.3934/mbe.2023396 |
[10] | Jean-Jacques Kengwoung-Keumo . Competition between a nonallelopathic phytoplankton and an allelopathic phytoplankton species under predation. Mathematical Biosciences and Engineering, 2016, 13(4): 787-812. doi: 10.3934/mbe.2016018 |
Many organisms exhibit cooperative behavior that benefits the whole group. Examples of such behavior in microbial populations are secreted products such as extracellular enzymes that digest nutrients, siderophores that acquire iron, or exopolysaccharides that support biofilm growth [16]. Such products are known as public goods. While common goods production is advantageous, and often critical to the growth of the population, it is also costly. Cheating individuals that choose not to cooperate could benefit from the fruits of labor of individuals that do, and would have a competitive advantage over the cooperators. One would expect that cheaters should be able to successfully invade a population of cooperators. This situation reflects a Prisoner's dilemma in evolutionary game theory, where cheating is the evolutionary stable strategy [12]. However, when cheaters start to dominate cooperators, the common good would become scarce, ultimately leading to the demise of cooperators and cheaters alike. This idea is known as the Tragedy of the Commons [8,6]. It has since then spurred a body of research that continues to grow up to this day, devoted to identifying mechanisms that explain the evolution of cooperation.
Although there has been extensive experimental work on the topic of the Tragedy of the Commons, perhaps somewhat surprisingly, there exist very few mathematical models based on first principles that express population growth, public good production and cheating behavior, for which the tragedy of the commons can be verified by means of a mathematical proof. In [11] we proposed a minimal chemostat model that incorporates cooperative and cheating behavior in microbial populations, and we established mathematically that the tragedy of the commons does indeed occur. The main goal of this paper is to generalize the chemostat model of [11] to investigate the fate of the population when the cooperator has the ability to evolve its ecological characteristics, specifically its per capita uptake rate function and yield constant. Although we find that the tragedy continues to hold in many cases, we also find scenarios where the cooperator not only survives, it even outcompetes the cheater by driving it to extinction. We will show that his happens when the break-even concentration for the growth nutrient of the cooperator evolves to become lower than that of the cheater. Interestingly, under no circumstances can the cooperator and cheater coexist in this generalized chemostat model: either the tragedy of the commons occurs, or the cooperator outcompetes the cheater, hereby favoring a very strong form of cooperation.
We propose the generalized chemostat model and prove our main result in Section 2. Section 3 contains a discussion of our main result, and conclusions are found in Section 4.
We consider a general chemostat model with positive dilution rate
S+E→S+P. |
The rate of this reaction is proportional to
As mentioned, the cooperator only uses a fraction of the available processed nutrient towards growth, and we denote this fraction by a positive constant
These considerations lead to the following model:
dSdt(t)=D(S0−S)−EG(S) | (2.1) |
dPdt(t)=EG(S)−1γ1X1F1(P)−1γ2X2F2(P)−DP | (2.2) |
dEdt(t)=η(1−q)X1F1(P)−DE | (2.3) |
dX1dt(t)=X1(qF1(P)−D) | (2.4) |
dX2dt(t)=X2(F2(P)−D) | (2.5) |
defined on the forward invariant set
By scaling
dsdt(t)=d(s0−s)−eg(s) | (2.6) |
dpdt(t)=eg(s)−x1f1(p)−x2f2(p)−dp | (2.7) |
dedt(t)=(1−q)x1f1(p)−de | (2.8) |
dx1dt(t)=x1(qf1(p)−d) | (2.9) |
dx2dt(t)=x2(f2(p)−d) | (2.10) |
Defining two new variables
m=s+p+e+x1+x2v=e−Qx1, where Q=1−qq, |
which satisfy the following equations:
dmdt(t)=d(s0−m)dvdt(t)=−dv, |
hence
dpdt(t)=Qx1g(s0−p−x1/q−x2)−x1f1(p)−x2f2(p)−dpdx1dt(t)=x1(qf1(p)−d)dx2dt(t)=x2(f2(p)−d), |
which is defined on the forward invariant state space
w=p+x1q+x2. |
instead of using the variable
dwdt(t)=Qx1g(s0−w)−dw | (2.11) |
dx1dt(t)=x1(qf1(w−x1/q−x2)−d) | (2.12) |
dx2dt(t)=x2(f2(w−x1/q−x2)−d) | (2.13) |
which is a system with forward invariant state space
(−Qx1g′−dQg0x1qf′1qf1−d−x1f′1−x1qf′1x2f′2−x2f′2/qf2−d−x2f′2), |
where we have suppressed the arguments of the functions
(∗+0+∗−+−∗) |
The key observation is that this sign structure implies that system
We begin our investigation of the dynamics of system
p1=a1dqm1−d. | (2.14) |
The quantity
H:p1<s0, |
which says that the break-even concentration of the cooperator should be less than the input nutrient concentration. If
h(w)=dQwg(s0−w). |
We showed in [11] that
h(w)=q(w−p1),0≤w<s0. | (2.15) |
Then strict convexity of the function on the left, and linearity of the function on the right, implies that this equation has at most
Lemma 2.1. Let
Figure 1 in the S3 Appendix in [11] illustrates the geometry of the nullclines. Using the fact that this planar system is cooperative and irreducible, one may conclude that
The dynamics of system
We are now ready to investigate the global behavior of system
Theorem 2.2. Assume that the conditions of Lemma
1. Strong cooperation or tragedy: If
2. Tragedy: If
Proof. Using Lemma
1. Suppose that
Jis=(−Qx1g′(s0−w)−dQg(s0−w)x1qf′1(w−x1/q)qf1(w−x1/q)−d−x1f′1(w−x1/q)), |
and the number
λi3=f2(w−x1/q)−d. |
Evaluating these at
We are left to show that all solutions converge to one of the 3 steady states. This follows from Lemma
Ωe={x1≥0,x2≥0,0≤w≤s0}, |
by extending the functions
As noted earlier, the time-reversed extending system
Consequently, we have
O⊂[A,B]K⊂int(Ωe). |
Then Proposition 4.3 in chapter 3 of [13] implies that
2. Suppose that
We are left to show that all solutions with
Having proved Theorem
Theorem 2.3. Assume that
1. Strong cooperation or tragedy: If
2. Tragedy: If
Figure 1 illustrates the case that
Theorem
The main purpose of this paper was to investigate whether a tragedy can be avoided if the cooperator evolves by either changing its yield constant
Therefore, the cooperator must modify
This has been observed during experimental evolution of cooperating microbial populations, where adaptations that improve nutrient uptake stabilize cooperative behavior [15,1]. More broadly, any principle or mechanism that disproportionally increases the benefit of cooperation in cooperating individuals, including kin selection or spatial structuring, will favor its evolution and maintenance [17].
Assuming that
Patrick De Leenheer is supported in part by NSF-DMS-1411853. Martin Schuster is supported by NSF-MCB-1158553. Hal Smith is supported by Simons Foundation Grant 355819.
All authors declare no conflicts of interest in this paper.
[1] | K.L. Asfahl, J. Walsh, K. Gilbert and M. Schuster, Non-social adaptation defers a tragedy of the commons in Pseudomonas aeruginosa quorum sensing, ISME J., 9 (2015), 1734–1746. |
[2] | J. S. Chuang, O. Rivoire and S. Leibler, Simpson's Paradox in a synthetic microbial system, Science, 323 (2009), 272–275. |
[3] | J. Cremer, A. Melbinger and E. Frey, Growth dynamics and the evolution of cooperation in microbial populations, Sci Rep, 2 (2012), 281. |
[4] | A.A. Dandekar, S. Chugani and E.P. Greenberg, Bacterial quorum sensing and metabolic incentives to cooperate, Science, 338 (2012), 264–266. |
[5] | F. Fiegna and G.J. Velicer, Competitive fates of bacterial social parasites: persistence and selfinduced extinction of Myxococcus xanthus cheaters, Proc. Biol. Sci., 270 (2003), 1527–1534. |
[6] | G.R. Hardin, The tragedy of the commons, Science, 162 (1968), 1243–1248. |
[7] | R. Kümmerli, A.S. Griffin, S. A.West, A. Buckling and F. Harrison, Viscous medium promotes cooperation in the pathogenic bacterium Pseudomonas aeruginosa, Proceedings of the Royal Society B, 276 (2009), 3531–3538. |
[8] | W.F. Lloyd, Two lectures on the checks to population (Oxford Univ. Press, Oxford, England, 1833), reprinted (in part, in: Population, Evolution, and Birth Control, G. Hardin, Ed. (Freeman, San Francisco, 1964), 37. |
[9] | T. Pfeiffer, S. Schuster and S. Bonhoeffer, Cooperation and competition in the evolution of ATPproducing pathways, Science, 292 (2001), 504–507. |
[10] | P.B. Rainey and K. Rainey, Evolution of cooperation and conflict in experimental bacterial populations, Nature, 425 (2003), 72–74. |
[11] | M. Schuster, E. Foxall, D. Finch, H. Smith and P. De Leenheer, Tragedy of the commons in the chemostat, PLOS ONE, Dec 2017. Available from: https://doi.org/10.1371/journal.pone.0186119. |
[12] | Introduction to Evolutionary Game Theory, Proceedings of Symposia in Applied Mathematics, 69 (2011), 1–25. |
[13] | H.L. Smith, Monotone Dynamical Systems, American Mathematical Society, 1995. |
[14] | H.L. Smith and P. Waltman, The theory of the chemostat, Cambridge University Press, 1994. |
[15] | A.J. Waite and W. Shou, Adaptation to a new environment allows cooperators to purge cheaters stochastically, PNAS, 109 (2012), 19079–19086. |
[16] | S.A. West, A.S. Griffin, A. Gardner and S.P. Diggle SP, Social evolution theory for microorganisms, Nat. Rev. Microbiol., 4 (2006), 597–607. |
[17] | S.A. West, A.S. Griffin and A. Gardner, Evolutionary explanations for cooperation, Curr. Biol. 17 (2007), R661–72. |
1. | Harry J. Gaebler, Hermann J. Eberl, Thermodynamic Inhibition in Chemostat Models, 2020, 82, 0092-8240, 10.1007/s11538-020-00758-3 | |
2. | Stephen R Lindemann, A piece of the pie: engineering microbiomes by exploiting division of labor in complex polysaccharide consumption, 2020, 30, 22113398, 96, 10.1016/j.coche.2020.08.004 | |
3. | Bryan K. Lynn, Patrick De Leenheer, Division of labor in bacterial populations, 2019, 316, 00255564, 108257, 10.1016/j.mbs.2019.108257 | |
4. | Constantinos Xenophontos, W. Stanley Harpole, Kirsten Küsel, Adam Thomas Clark, Cheating Promotes Coexistence in a Two-Species One-Substrate Culture Model, 2022, 9, 2296-701X, 10.3389/fevo.2021.786006 | |
5. | Bryan K. Lynn, Patrick De Leenheer, Martin Schuster, Bashir Sajo Mienda, Putting theory to the test: An integrated computational/experimental chemostat model of the tragedy of the commons, 2024, 19, 1932-6203, e0300887, 10.1371/journal.pone.0300887 |