## Abstract

The idea that selection works in different ways during free population growth and at the equilibrium population size has been present in theoretical biology for a long time. It was first expressed as an *r* and *K* selection concept and later clarified in the debate on fitness measures in life history theory. The latest discussion related to this topic is focused on the nest site lottery mechanism and the resulting new population growth model. In this mechanistic biphasic model, the suppression of growth is induced by a shortage of free nest sites for newborns. Before it occurs, the population can grow exponentially. In this paper, the continuous version of the model and its selective properties are analysed. We show a continuous smooth transition between different fitness measures operating during the exponential growth and suppressed growth phase and at the equilibrium population size. Then, the model is extended to the case of a population of parasites, where a constant number of nest sites is replaced by the dynamics of a population of their hosts, in the role of the limiting supply. Parasite strategies are selected under exponential and suppressed growth phases of the population of hosts. Transitions between different fitness measures and conditions for extinction of hosts by parasites are analysed. An interesting result is the possibility of a continuum of fitness measures of parasites for the unsuppressed exponential growth of the host population.

This is a preview of subscription content, access via your institution.

## Change history

### 10 February 2020

Unfortunately, part of the article title was updated as subtitle which in turn resulted with complete title not appearing on website and in the bibliographic data. The complete version of title is updated here.

## References

Argasinski K, Broom M (2013a) Ecological theatre and the evolutionary game: how environmental and demographic factors determine payoffs in evolutionary games. J Math Biol 67:935–962

Argasinski K, Broom M (2013b) The nest site lottery: how selectively neutral density dependent growth suppression induces frequency dependent selection. Theor Popul Biol 90:82–90

Argasinski K, Broom M (2018a) Evolutionary stability under limited population growth: eco-evolutionary feedbacks and replicator dynamics. Ecol Complex 34:198–212

Argasinski K, Broom M (2018b) Interaction rates, vital rates, background fitness and replicator dynamics: how to embed evolutionary game structure into realistic population dynamics. Theory Biosci 137:33–50

Argasinski K, Rudnicki R (2017) Nest site lottery revisited: towards a mechanistic model of population growth suppressed by the availability of nest sites. J Theor Biol 420:279–289

Barbault R (1987) Are still r-selection and K-selection operative concepts? Acta Oecol 8:63–70

Carlson SM, Quinn TP, Hendry AP (2011) Eco-evolutionary dynamics in Pacific salmon. Heredity 106:438–447

Cressman R, Garay J (2003) Evolutionary stability in Lotka–Volterra systems. J Theor Biol 222:233–245

Cressman R, Garay J, Hofbauer J (2001) Evolutionary stability concepts for N-species frequency-dependent interactions. J Theor Biol 211:1–10

Dańko MJ, Burger O, Kozłowski J (2017) Density-dependence interacts with extrinsic mortality in shaping life histories. PLoS ONE 12:e0186661

Dańko A, Schaible R, Pijanowska J, Dańko MJ (2018a) Population density shapes patterns of survival and reproduction in

*Eleutheria dichotoma*(Hydrozoa: Anthoathecata). Mar Biol 165:48Dańko MJ, Burger O, Argasinski K, Kozłowski J (2018b) Extrinsic mortality can shape life-history traits, including senescence. Evolut Biol 45:395–404

Doebeli M, Ispolatov Y, Simon B (2017) Point of view: towards a mechanistic foundation of evolutionary theory. eLIFE 6:e23804

Engen S, Saether BE (2017) r-and K-selection in fluctuating populations is determined by the evolutionary trade-off between two fitness measures: growth rate and lifetime reproductive success. Evolution 71(1):167–173

Ferriere R, Legendre S (2013) Eco-evolutionary feedbacks, adaptive dynamics and evolutionary rescue theory. Philos Trans R Soc Lond B Biol Sci 368:20120081

Garay J, Csiszár V, Móri TF (2017) Survival phenotype, selfish individual versus Darwinian phenotype. J Theor Biol 430:86–91

Geritz S, Kisdi E (2012) Mathematical ecology: why mechanistic models? J Math Biol 65:1411–1415

Grimm V, Railsback SFS (2005) Individual-based modeling and ecology. Princeton series in theoretical and computational biology. Princeton University Press, Princeton

Hendry AP (2016) Eco-evolutionary dynamics. Princeton university press

Hui C (2006) Carrying capacity, population equilibrium, and environment’s maximal load. Ecol Model 192:317–320

Hui C (2015) International encyclopedia of the social & behavioral sciences, vol 3. Elsevier, Amsterdam

Hutchinson GE (1965) The ecological theater and the evolutionary play. Yale University Press, New Haven

Kozłowski J (1980) Density dependence, the logistic equation, and r- and K-selection: a critique and an alternative approach. Evolut Theory 5:89–101

Kozłowski J (1993) Measuring fitness in life history studies. Trends Ecol Evol 8:84–85

Kozłowski J (2006) Why life histories are diverse. Pol J Ecol 54:585–605

Lion S, Metz JA (2018) Beyond R\(_0\) maximisation: on pathogen evolution and environmental dimensions. Trends Ecol Evol 33:458–473

Łomnicki A (1988) Population ecology of individuals. Princeton University Press, Princeton

Maynard Smith J (1982) Evolution and the theory of games. Cambridge University Press, Cambridge

Metz JA, Nisbet RM, Geritz SA (1992) How should we define ‘fitness’ for general ecological scenarios? Trends Ecol Evol 7:198–202

Metz J, Mylius S, Diekmann O (2008a) Even in the odd cases when evolution optimizes, unrelated population dynamical details may shine through in the ESS. Evolut Ecol Res 10:655–666

Metz J, Mylius S, Diekmann O (2008b) When does evolution optimize? Evolut Ecol Res 10:629–654

Pianka ER (1970) On r-and k-selection. Am Nat 104:592–597

Post DDM, Palkovacs EPE (2009) Eco-evolutionary feedbacks in community and ecosystem ecology: interactions between the ecological theatre and the evolutionary play. Philos Trans R Soc Lond B Biol Sci 364:1629–1640

Reznick D, Bryant MJ, Bashey F (2002) r-and K-selection revisited: the role of population regulation in life-history evolution. Ecology 83(6):1509–1520

Roff D (1992) Evolution of life histories: theory and analysis. Springer, Berlin

Rudnicki R (2017) Does a population with the highest turnover coefficient win competition? J Differ Equ Appl 23:1529–1541

Saether BE, Visser ME, Grřtan V, Engen S (2016) Evidence for r-and K-selection in a wild bird population: a reciprocal link between ecology and evolution. Proc R Soc B Biol Sci 283(1829):20152411

Stearns S (1992) The evolution of life histories. Oxford University Press, Oxford

Traulsen A, Claussen JC, Hauert C (2005) Coevolutionary dynamics: from finite to infinite populations. Phys Rev Lett 95:1–4

Uchmański J, Grimm V (1996) Individual-based modelling in ecology: what makes the difference? Trends Ecol Evol 11(10):437–441

## Acknowledgements

We want to thank Mark Broom, Jan Kozłowski and John McNamara for their support of the project and helpful suggestions. In addition, we want to thank anonymous reviewer for valuable comments and suggestions. This paper was supported by the Polish National Science Centre Grant No. 2013/08/S/NZ8/00821 FUGA2 (KA) and Grant No. 2017/27/B/ST1/00100 OPUS (RR).

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The original version of this article was revised: Part of title was updated as sub-title in the original publication and corrected version of title is updated here.

## Appendices

### Appendix 1: Proof of Theorem 1

Let us start from subsystem (29)

Let us substitute \(\left( \dfrac{K}{n(t)}-1\right) /\tau ={\bar{d}}(t)+\alpha\) , which implies \(\alpha =\dfrac{K-n(t)}{n(t)\tau }-{\bar{d}}(t)\). This leads to

The last bracketed term can be presented in the following form:

Let us substitute the density attractor \({\tilde{n}}(t)=\dfrac{K}{\tau{\bar{d}} (t)+1 }\) in the above formula. This will lead to the form being a function of the distance from the stable density manifold measured by the bracketed term:

In effect, we obtain the following system:

where the factor

is compatible with the nest site lottery mechanism (which implies maximization of \(L_{i}=b_{i}/d_{i}\) and \(d_{i}\) among strategies with maximal \(L_{i}\)). Let us consider the second factor

where the term \(\left( \dfrac{b_{i}}{{\bar{b}}(t)}-1\right)\), responsible for the maximization of \(b_{i}\), and the term \(\left( \dfrac{{\tilde{n}}(t)}{n(t)} -1\right)\) vanish with convergence to the nullcline \({\tilde{n}}(t)\). After the substitution of the switching point \({\hat{n}}=\dfrac{K}{\tau {\bar{b}}+1}\) as the population size, the replicator dynamics (29) reduce to the unsuppressed case \({\dot{q}}_{i}(t)=(b_{i}-{\bar{b}}(t))-\left( d_{i}-{\bar{d}} (t)\right)\). This means that the decrease in the coefficient

is responsible for the continuous transition from unsuppressed exponential growth to the nest site lottery mechanism.

### Appendix 2: Proof of Theorem 2

Thus, the function *x*(*t*) is a solution of the differential equation \({\dot{x}}(t)=P(x(t))\), where \(P(x)=\tau ^{-1}-ax+D_{p}x^{2}\) and \(a=\tau ^{-1}+d+D_{p}+B\). Note that in the above derivation, *D* cancels out. Therefore, every other constant or function added to *D* will also vanish. This means that *x* is independent of all host mortality factors. Since

we find that \(\Delta >0\), and the quadratic polynomial *P*(*x*) has two positive roots:

The points \(x_{*}\) and \(x^{*}\) are, respectively, stable and unstable stationary solutions of the equation \({\dot{x}}(t)=P(x(t))\). We have \(x_{*}<1\). Indeed, \(x_{*}<1\) if \(a-2D_{p}<\sqrt{ a^{2}-4D_{p}\tau ^{-1}}\), leading to \(a>D_{p}+\tau ^{-1}\), which is obviously always satisfied. Furthermore, from \(a>D_{p}+\tau ^{-1}\), we find that \(2D_{p}-a<\sqrt{a^{2}-4D_{p}\tau ^{-1}}\), which yields \(x^{*}>1\). Recall that by definition, \(n\le K\); thus, we have that \(x\le 1\). Therefore, \(x^{*}\) can be rejected as nonbiological and *x*(*t*) converges to \(x_{*}\) as \(t\rightarrow \infty\).

### Appendix 3: Proof of Theorem 3

We know that trajectories can converge to (0, 0) or escape to infinity. System (38)–(40) can be presented in the form

If condition (42) is satisfied, then we enter the suppression phase when \(n(t)\tau b>K(t)-n(t)\) and remain there. This means that for a sufficiently large time, we have the system

The last system is linear, and the behaviour of its solutions depends on the eigenvalues of the matrix

The eigenvalues of *A* satisfy the following equations:

equivalent to

We have

This means that the eigenvalues of *A* are real numbers and the solutions of (62)–(63) are bounded if and only if \(\lambda _{1}\le 0\) and \(\lambda _{2}\le 0\), which holds if

The last inequalities are equivalent to the following:

i.e.,

This is the condition when the parasites “kill” the hosts, since the only stationary point is (0, 0) , meaning total extinction.

### Appendix 4: Proof of Theorem 4

From “Appendix 2”, we have that the dynamics of \(x(t)=n(t)/K(t)\) are described by equation \({\dot{x}}(t)=\tau ^{-1}-a(t)x(t)+D_{p}x^{2}(t)\). Since the right-hand side is quadratic with two positive roots \(x_{*}(t)<1\) and \(x^{*}(t)>1\) (see “Appendix 2”), it can be presented in the form of the product of the roots. Describing the dynamics of *x*(*t*) as

directly leads to (49). From (58) in “Appendix 1”, we have that the dynamics of strategy frequencies (45) can be presented in the form

where the last bracketed term \(\left( \dfrac{K(t)-n(t)}{n(t)\tau }-{\bar{d}} (t)\right)\) can be expressed in terms of \(x(t)=n(t)/K(t)\), leading to (48).

### Appendix 5: Proof of Theorem 5

Now, we should calculate the nullclines of the extended equation and their stability. The manifolds where the right-hand side of (51) equals zero will satisfy the equation

obtained from the bracketed term of (51). Here,

and \(R>2\sqrt{D_{d}D_{p}n}\) is the condition for the existence of two nullclines:

where \(K_{1}\) is an unstable extinction barrier and \(K_{2}\) is the attracting nullcline. Since the additional factor \(D_{d}K\) cannot affect Theorem 2, we can apply it and change coordinates in the system (50)–(51) from *n*, *K* to *x*, *K*. In effect, we obtain the system

Then, Eq. (65) has the attracting nullcline \({\tilde{K}}=\dfrac{ R-D_{p}x}{D_{d}}\), and Eq. (64) converges to \(x_{*}\). Combining them leads to equilibrium \(x_{*}\) and \({\hat{K}}=\dfrac{ R-D_{p}x_{*}}{D_{d}}\), which is positive for \(R>D_{p}x_{*}\).

### Appendix 6: Proof of Theorem 6

Note that in the system (55)–(57), in addition to the strategy specific parasite mortalities \(d_{i}\), we have additional mortalities caused by the death of the host equal to \(D-D_{p}-D_{d}K(t)\). We can incorporate them into aggregated mortalities \(d_{i}^{A}(K(t))=d_{i}+D+D_{p}+D_{d}K(t)\) (and thus, \({\bar{d}}^{A}(t,K(t))={\bar{d}} (t)+D+D_{p}+D_{d}K(t)\)). Then, analogously to (58) from “Appendix 1” and (48), the frequency dynamics can be presented as

On the nullcline \({\tilde{n}}(t,K(t))\) (52), where the bracketed term of the r.h.s of the parasite population size equation (56) is zero, we have that \({\tilde{x}}(t,K(t))={\tilde{n}}(t,K(t))/K(t)\), and in effect, we obtain

and this factor is an increasing function of *K*. Then, the frequency dynamics can be presented as

Recall the proof of Theorem 2. Therefore, we can replace Eq. (56) by the equation

as in Theorems 4 and 5. Then, the factor \(\dfrac{1}{ x(t)\tau }\) will be attracted by \(\dfrac{1}{x_{*}(t)\tau }\). Similar to Theorem 5, Eq. (57) can be denoted in terms of *x*(*t*) as

Thus, the density subsystem (56)–(57) is attracted by \(x_{*}(t)\) and \({\hat{K}}(t)=\dfrac{R-D_{p}x_{*}(t)}{D_{d}}\), and \(R\le D_{p}x_{*}(t)\) is the condition for evolutionary suicide.

Now, fix the value of *q*(*t*) and focus on the subspace *n*, *K* (which can be presented as *x*, *K*). For each particular point *q*, we have a specific value of \({\bar{d}}(q)\). From Theorem 5, we know that for every fixed value of *d*, a unique stable equilibrium \(x_{*}\) exists, which determines the respective stable value of \({\hat{K}}\). Since this is a stationary point, it should be the intersection of the attracting nullcline \({\tilde{n}}(t,K)\) and the attracting nullcline \({\tilde{K}}_{2}\) (54) from Theorem 5. Therefore, at this point, we have that \(x_{*}={\tilde{x}}(t,{\hat{K}})\) and in effect

For every value of *q* from the strategy simplex, we can define point \(x_{*}\), and for that point, we can define the respective \({\hat{K}}\). Both \(x_{*}\) and \({\hat{K}}\) constitute, respectively, the attracting surfaces for system (56, 57). Since \({\hat{K}}\) depends only on \(x_{*}\), which in turn depends on the value of \({\bar{d}}\) determined by the strategic composition of the population *q*(*t*), these attracting surfaces can be described as

On the surface \(\left( \breve{x}_{*}(q),\breve{K}(q)\right)\), condition (66) is always satisfied, leading to the pure nest site lottery mechanism operating there.

## Rights and permissions

## About this article

### Cite this article

Argasinski, K., Rudnicki, R. From nest site lottery to host lottery: continuous model of growth suppression driven by the availability of nest sites for newborns or hosts for parasites and its impact on the selection of life history strategies.
*Theory Biosci.* **139**, 171–188 (2020). https://doi.org/10.1007/s12064-019-00307-0

Received:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s12064-019-00307-0

### Keywords

- Eco-evolutionary feedback
- Carrying capacity
- Density dependence
- Frequency-dependent selection
- Fitness measures
*r*and*K*selection*R*_{0}and*r*maximization