Hierarchy Establishment from Nonlinear Social Interactions and Metabolic Costs: An Application to Harpegnathos saltator
Abstract
:1. Introduction
2. Model Formulation
- (a)
- Modeling Relative Cost: Assuming that gamergates are the only members of the colony who actively reproduce, the cost for all gamergates of remaining in their status in the colony is measured by (Kcal) which consists of the energy cost of laying eggs as well as the energy to perform behaviors that maintain the social hierarchy such as antennal dueling and dominance behaviors. According to the ecological metabolism theory, the formulation of is modeled byApplying the similar arguments to non-reproductive workers , their relative cost is defined asNote that
- (b)
- Colony Size N: Let e be the average number of eggs produced per gamergate, then the total number of eggs produced at time t would be . The survival of eggs is directly linked to the care provided by the number of non-reproductive workers [32]. Thus, we assume that the survival rate of eggs is proportional to the relative cost of the worker class . Therefore, the input of new members is obtained as followsWe assume that the colony has a density-dependent mortality rate due to limited resources, given by
- (c)
- Gamergates and Workers : We adopt the concept of [15] to model that a non-reproductive worker would leave its worker status and enter into the gamergate phase when the worker’s relative cost is higher than the gamergate’s relative cost , and vice versa [15,32]. Since it has been shown that gamergates can revert to worker status [31] and non-reproductive workers can become gamergates [33], we assume that an ant can interchange between a gamergate and non-reproductive worker status based on the relative cost of remaining in a particular group. This relative cost derives from metabolic costs and is also related to its contribution to colony function. Thus, the population dynamics of gamergates is modeled as followsTo model the non-reproductive workers’ population dynamics, we assume that all newborns are in the non-reproductive worker status first, with the possibility of transitioning into the status of gamergates. We apply a similar approach as above so that the population dynamics of non-reproductive workers is given by
- What are the key life-history parameters that determine the dynamics of the different classes in the colony ?
- How do the metabolic parameters, such as , and the exponent parameters affect the colony size, N, and the size of the reproductive class ?
- How do the colony size, N, and the number of established gamergates, , regulate each other?
3. Mathematical Analysis
- 1.
- If , then there exists such that and has a unique solution in and no solution in .
- 2.
- If , has a unique solution in and no solution in .
- 3.
- If , then there exists such that and has no solution in . In , we have the following statements:
- (a)
- If , has at least one positive solution.
- (b)
- If , has no positive solution provided that is small enough.
- (c)
- If , has at least one positive solution provided that , while no positive solution provided that is small enough.
- The values of and are determined by the average reproduction ability of gamergate e and the average mortality of gamergates and workers, which provide an upper bound for the reproductive ratio.
- Note that can be rewritten asis increasing as increases. If , we also have that is increasing as increases. One potential biological meaning of this result is that when the mortality of the non-reproductive workers is sufficiently close but larger than that of the gamergates, increasing the mortality of the non-reproductive workers would cause a higher proportion of gamergates in the colony since they would reproduce more to make up for the loss of non-reproductive workers.
- Note that can be rewritten asOne potential biological meaning of this result is that when the mortality of non-reproductive workers is much larger than that of the gamergates, decreasing the mortality of gamergates would result in a smaller proportion of gamergates as they would reproduce more during its life time.
4. Model Validations and Simulations
4.1. Data and Parameter Estimations
4.2. Simulations and Bifurcations
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. Mathematical Proofs
Appendix A.1. Proof of Theorem 1
- . Let , thenSolving the above differential equation gives
- , thenSolving the above differential equation gives
- . Let , thenSolving the above differential equation gives
Appendix A.2. Proof of Theorem 2
- , is increasing in the both intervals of and , and for and for ;
- .
- , , is increasing in the both intervals of and , and for and for ;
- .
- When , we can get , and henceNotice that , and is increasing from zero to infinity in , we can conclude that has at least one positive solution in .
- When , we can get . Due toLetThen, if is large such that , we haveNotice that . Thus, if is small enough, then , and hence has no solution in .
- When , we have . Since is increasing in from zero to infinity while is increasing in from zero to infinity, we can get that has at least one positive solution in if , i.e., . If , then similar to the arguments for case , we can prove that has no solution in if is small enough.
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Parameter | Description | Best Estimation | Range | Units | Reference |
---|---|---|---|---|---|
e | Egg-laying rate | 2.42 | eggs/day | Estimated | |
Natural death rate of gamergates | 0.00028 | [27] | |||
Natural death rate of workers | 0.0025 | [27] | |||
Normalized metabolic rates for gamergates | 33.00 | Assumed | |||
Normalized metabolic rates for workers | 74.60 | Assumed | |||
Hypometric relation factor of gamergates | 0.99 | 1 | Assumed | ||
Hypometric relation factor of workers | 0.261 | 1 | Assumed |
Colony | Day | Colony Size (N) | Number of Gamergates () | Reproductive Ratio (X) |
---|---|---|---|---|
F44 | 0 | 42 | 3 | 0.0714 |
94 | 41 | 3 | 0.0732 | |
613 | 62 | 7 | 0.1129 | |
SAFC12 | 0 | 127 | 4 | 0.0315 |
94 | 109 | 4 | 0.0360 | |
125 | 91 | 4 | 0.0440 |
Colony | ||||||
---|---|---|---|---|---|---|
F44 | 92.77 | 1.61 | 0.37 | 51 | 5 | 0.098 |
SAFC12 | 121.15 | 1.59 | 0.67 | 97 | 15 | 0.155 |
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Bustamante-Orellana, C.; Bai, D.; Cevallos-Chavez, J.; Kang, Y.; Pyenson, B.; Xie, C. Hierarchy Establishment from Nonlinear Social Interactions and Metabolic Costs: An Application to Harpegnathos saltator. Appl. Sci. 2022, 12, 4239. https://doi.org/10.3390/app12094239
Bustamante-Orellana C, Bai D, Cevallos-Chavez J, Kang Y, Pyenson B, Xie C. Hierarchy Establishment from Nonlinear Social Interactions and Metabolic Costs: An Application to Harpegnathos saltator. Applied Sciences. 2022; 12(9):4239. https://doi.org/10.3390/app12094239
Chicago/Turabian StyleBustamante-Orellana, Carlos, Dingyong Bai, Jordy Cevallos-Chavez, Yun Kang, Benjamin Pyenson, and Congbo Xie. 2022. "Hierarchy Establishment from Nonlinear Social Interactions and Metabolic Costs: An Application to Harpegnathos saltator" Applied Sciences 12, no. 9: 4239. https://doi.org/10.3390/app12094239
APA StyleBustamante-Orellana, C., Bai, D., Cevallos-Chavez, J., Kang, Y., Pyenson, B., & Xie, C. (2022). Hierarchy Establishment from Nonlinear Social Interactions and Metabolic Costs: An Application to Harpegnathos saltator. Applied Sciences, 12(9), 4239. https://doi.org/10.3390/app12094239