# Coexistence and Replacement of Two Different Maturation Strategies Adopted by a Stage-Structured Population

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Two-Strategy Two-Stage Model

#### 2.1. Existence of the Equilibria

**Theorem**

**1.**

- 1.
- There is always a trivial equilibrium ${E}_{0}=(0,0,0,0)$ in any situation.
- 2.
- If ${R}_{{S}_{1}}>1$, there exists a single-strategy boundary equilibrium$${E}_{1}=({D}_{A}\left({S}_{1}\right){A}_{1}^{*}/{S}_{1},K(1-1/{R}_{{S}_{1}}),0,0).$$
- 3.
- If ${R}_{{S}_{2}}>1$, there exists a single-strategy boundary equilibrium$${E}_{2}=(0,0,{D}_{A}\left({S}_{2}\right){A}_{2}^{*}/{S}_{2},K(1-1/{R}_{{S}_{2}})).$$
- 4.
- If the condition ${R}_{{S}_{1}}={R}_{{S}_{2}}>1$ holds, there exists a cluster of positive equilibria$${E}_{p}=\left(\right)open="("\; close=")">{J}_{1}^{*},{A}_{1}^{*},{J}_{2}^{*},{A}_{2}^{*}\left)=\right({\displaystyle \frac{{D}_{A}\left({S}_{1}\right){A}_{1}^{*}}{{S}_{1}}},{\displaystyle \frac{K(1-1/{R}_{{S}_{1}})}{1+c}},{\displaystyle \frac{{D}_{A}\left({S}_{2}\right){A}_{2}^{*}}{{S}_{2}}},{\displaystyle \frac{cK(1-1/{R}_{{S}_{2}})}{1+c}}$$

#### 2.2. Linear Stability Analysis of the Four Equilibria

**Theorem**

**2.**

- 1.
- If ${R}_{{S}_{1}}<1$ and ${R}_{{S}_{2}}<1$, ${E}_{0}=(0,0,0,0)$ is always locally asymptotically stable;
- 2.
- If ${R}_{{S}_{1}}>1,{R}_{0}>1$, the boundary equilibrium ${E}_{1}=({J}_{1}^{*},{A}_{1}^{*},0,0)$ is locally asymptotically stable;
- 3.
- If ${R}_{{S}_{2}}>1,{R}_{0}<1$, the boundary equilibrium ${E}_{2}=(0,0,{J}_{2}^{*},{A}_{2}^{*})$ is locally asymptotically stable.

#### 2.3. The Global Stability of the Boundary Equilibria

**Theorem**

**3.**

- 1.
- If ${R}_{{S}_{1}}>1>{R}_{{S}_{2}}$, the boundary equilibrium ${E}_{1}=({J}_{1}^{*},{A}_{1}^{*},0,0)$ is globally asymptotically stable;
- 2.
- If ${R}_{{S}_{2}}>1>{R}_{{S}_{1}}$, the boundary equilibrium ${E}_{2}=(0,0,{J}_{2}^{*},{A}_{2}^{*})$ is globally asymptotically stable.

#### 2.4. Numerical Results

## 3. The Adaptive Dynamical System

**Theorem**

**4.**

## 4. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Gemmill, A.W.; Skorping, A.; Read, A.F. Optimal timing of first reproduction in parasitic nematodes. J. Evol. Biol.
**1999**, 12, 1148–1156. [Google Scholar] [CrossRef] - Brommer, J.E. The evolution of fitness in life-history theory. Biol. Rev. Camb. Philos. Soc.
**2000**, 75, 377–404. [Google Scholar] [CrossRef] [PubMed] - Roff, D.A. The Evolution of Life Histories: Theory and Analysis; Chapman and Hall: New York, NY, USA, 1992. [Google Scholar]
- Stearns, S.C. The Evolution of Life Histories; Oxford University Press: Oxford, UK, 1992. [Google Scholar]
- Fujiwara, M.; Pfeiffer, G.; Boggess, M.; Day, S.; Walton, J. Coexistence of Competing Stage-Structured Populations. Sci. Rep.
**2011**, 1, 107. [Google Scholar] [CrossRef] - Beverton, R.J.H.; Holt, S.J. On the Dynamics of Exploited Fish Populations; Ministry of Agriculture, Fisheries and Food: London, UK, 1957. [Google Scholar]
- Moll, J.D.; Brown, J.S. Competition and coexistence with multiple life-history stages. Am. Nat.
**2008**, 171, 839–843. [Google Scholar] [CrossRef] [PubMed] - Tian, Y.; Liu, X. Adaptive evolution of life history strategies related to maturation time in seasonal environment. Ecol. Complex.
**2019**, 40, 100794. [Google Scholar] [CrossRef] - Gadgilm, M.; Bossert, W.H. Life historical consequences of natural selection. Am. Nat.
**1970**, 104, 1–24. [Google Scholar] [CrossRef] - Williams, G.C. The costs of reproduction; a refinement of Lack’s principle, in Natural selection. Am. Nat.
**1966**, 100, 687–690. [Google Scholar] [CrossRef] - Law, R. Optimal life histories under age-specific predation. Am. Nat.
**1979**, 114, 399–417. [Google Scholar] [CrossRef] - Hutchings, J.A. The influence of growth and survival costs of reproduction on Atlantic cod, gadus morhua, population growth rate. Can. J. Fish. Aquat. Sci.
**1999**, 56, 1612–1623. [Google Scholar] [CrossRef] - Laurian, C.; Ouellet, J.P.; Courtois, R.; Breton, L.; St-Onge, S. Effects of intensive harvesting on moose reproduction. J. Appl. Ecol.
**2000**, 37, 515–531. [Google Scholar] [CrossRef] - Ernande, B.; Dieckmann, U.; Heino, M. Adaptive changes in harvested populations: Plasticity and evolution of age and size at maturation. Proc. R. Soc. Lond. B
**2004**, 271, 415–423. [Google Scholar] [CrossRef] [PubMed] - Xue, S.; Li, M.; Ma, J.; Li, J. The effect of harvesting adults on the evolution of reproduction age via density-dependent juvenile mortality. Bull. Math. Biol.
**2021**, 83, 108. [Google Scholar] [CrossRef] [PubMed] - Tsurim, I.; Silberbush, A.; Ovadia, O.; Blaustein, L.; Margalith, Y. Inter- and Intra-Specific Density-Dependent Effects on Life History and Development Strategies of Larval Mosquitoes. PLoS ONE
**2012**, 8, e57875. [Google Scholar] [CrossRef] [PubMed] - Cai, L.; Ai, S.; Li, J. Dynamics of mosquitoes populations with different strategies for releasing sterile mosquitoes. SIAM J. Appl. Math.
**2014**, 74, 1786–1809. [Google Scholar] [CrossRef] - Neverova, G.P.; Zhdanova, O.L.; Ghosh, B.; Frisman, E.Y. Dynamics of a discrete-time stage-structured predator-prey system with Holling type II response function. Nonlinear Dyn.
**2019**, 98, 427–446. [Google Scholar] [CrossRef] - Pal, D.; Ghosh, B.; Kar, T.K. Hydra effects in stable food chain models. BioSystems
**2019**, 185, 104018. [Google Scholar] [CrossRef] [PubMed] - Perko, L. Differential Equations and Dynamical Systems. In Texts in Applied Mathematics, 3rd ed.; Springer: Berlin/Heidelberg, Germany, 2001; Volume 7. [Google Scholar]
- Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, MA, USA, 1982. [Google Scholar]
- Christiansen, F.B. On conditions for evolutionary stability for a continuously varying character. Am. Nat.
**1991**, 138, 37–50. [Google Scholar] [CrossRef] - Geritz, S.A.; Kisdi, E.; Meszéna, G.; Metz, J.A. Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol.
**1998**, 12, 35–57. [Google Scholar] [CrossRef] - Eshel, I. Evolutionary and continuous stability. J. Math. Biol.
**1983**, 34, 485–510. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Flow chart of two-strategy two-stage model. Juveniles ${J}_{i}$ mature into adults ${A}_{i}$ at maturity rate ${S}_{i}$, $i=1,2$. Adults ${A}_{i}$ reproduce new offspring in the juvenile stage at per capita birth rate $B\left({S}_{i}\right)$. The death rates of juveniles and adults are ${D}_{J}\left({S}_{i}\right)$ and ${D}_{A}\left({S}_{i}\right)$, respectively.

**Figure 3.**Partition area according to the stability conditions of the equilibrium. I Region: only ${E}_{0}$ exists and is globally asymptotically stable; II Region: only ${E}_{1}$ is locally asymptotically stable; III Region: only ${E}_{2}$ is locally asymptotically stable; The solid line between II Region and III Region: only ${E}_{p}$ exist, but the stability is unclear because of the existence of a zero eigenvalue.

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**MDPI and ACS Style**

Xue, S.
Coexistence and Replacement of Two Different Maturation Strategies Adopted by a Stage-Structured Population. *Mathematics* **2023**, *11*, 2393.
https://doi.org/10.3390/math11102393

**AMA Style**

Xue S.
Coexistence and Replacement of Two Different Maturation Strategies Adopted by a Stage-Structured Population. *Mathematics*. 2023; 11(10):2393.
https://doi.org/10.3390/math11102393

**Chicago/Turabian Style**

Xue, Shuyang.
2023. "Coexistence and Replacement of Two Different Maturation Strategies Adopted by a Stage-Structured Population" *Mathematics* 11, no. 10: 2393.
https://doi.org/10.3390/math11102393