The effect of fight cost structure on fighting behaviour involving simultaneous decisions and variable investment levels

Mark Broom; Michal Johanis; Jan Rychtář

doi:10.1007/s00285-017-1149-y

Journal of Mathematical Biology

January 2018, Volume 76, Issue 1–2, pp 457–482 | Cite as

The effect of fight cost structure on fighting behaviour involving simultaneous decisions and variable investment levels

Authors
Authors and affiliations

Mark Broom
Michal Johanis
Jan RychtářEmail author

Article

First Online: 21 June 2017

Received: 21 January 2017

Revised: 30 May 2017

Abstract

In the “producer–scrounger” model, a producer discovers a resource and is in turn discovered by a second individual, the scrounger, who attempts to steal it. This resource can be food or a territory, and in some situations, potentially divisible. In a previous paper we considered a producer and scrounger competing for an indivisible resource, where each individual could choose the level of energy that they would invest in the contest. The higher the investment, the higher the probability of success, but also the higher the costs incurred in the contest. In that paper decisions were sequential with the scrounger choosing their strategy before the producer. In this paper we consider a version of the game where decisions are made simultaneously. For the same cost functions as before, we analyse this case in detail, and then make comparisons between the two cases. Finally we discuss some real examples with potentially variable and asymmetric energetic investments, including intraspecific contests amongst spiders and amongst parasitoid wasps. In the case of the spiders, detailed estimates of energetic expenditure are available which demonstrate the asymmetric values assumed in our models. For the wasps the value of the resource can affect the probabilities of success of the defender and attacker, and differential energetic investment can be inferred. In general for real populations energy usage varies markedly depending upon crucial parameters extrinsic to the individual such as resource value and intrinsic ones such as age, and is thus an important factor to consider when modelling.

Keywords

Kleptoparasitism Food stealing Producer–scrounger Game theory Simultaneous decisions

Mathematics Subject Classification

91A05 92D50

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Notes

Acknowledgements

M. Johanis was supported by the grant GAČR and GAČR 16-07378S. J. Rychtář was supported by the Simons Foundation Grant 245400.

Appendix

Table 2

Conditions for all types of equilibria. Here, we give conditions on parameters \(S_1\), \(S_2\), \(P_1\), and \(P_2\) under which equilibria are present for some value of v

#	Equilibrium	Conditions on \(S_1\), \(S_2\), \(P_1\), \(P_2\)
1	(0, 0)	Always possible
2	\((S_1,0)\)	For \(\alpha \ge 1\) always possible, for \(\alpha <1\) if and only if
2	\((S_1,0)\)	\(\frac{P_1}{S_1}\ge c>0\), where \((1+c)^\alpha =1+\frac{c}{1+c}\)
3	\((S_1,P_1)\)	For \(\alpha \ge 1\) always possible, for \(\alpha <1\) if and only if
3	\((S_1,P_1)\)	\(\frac{(1-\alpha )^{1/\alpha }}{1-(1-\alpha )^{1/\alpha }}\le \frac{P_1}{S_1}\le \frac{1-(1-\alpha )^{1/\alpha }}{(1-\alpha )^{1/\alpha }}\)
4	\((S_1,f(S_1))\)	\(P_1\le S_1\) and moreover \(\frac{S_1}{P_2}\le \frac{1-(1-\alpha )^{1/\alpha }}{(1-\alpha )^{1/\alpha }}\) if \(\alpha <1\)
5	\((f(P_1),P_1)\)	\(S_1\le P_1\) and moreover \(\frac{P_1}{S_2}\le \frac{1-(1-\alpha )^{1/\alpha }}{(1-\alpha )^{1/\alpha }}\) if \(\alpha <1\)
6	\((S_1,P_2)\)	\(P_2\le S_1\)
7	\((S_2,P_1)\)	\(S_2\le P_1\)
8	\(\Bigl (\frac{1}{2}\bigl (\frac{v}{2\alpha }\bigr )^{\frac{1}{\alpha }},\frac{1}{2}\bigl (\frac{v}{2\alpha }\bigr )^{\frac{1}{\alpha }}\Bigr )\)	\(P_1\le S_2\) and \(S_1\le P_2\)
9	\((f(P_2),P_2)\)	\(P_2\le S_2\)
10	\((S_2,f(S_2))\)	\(S_2\le P_2\)
11	\((S_2,P_2)\)	Always possible

Here we consider alternative ways of representing some of the results from the main body of the paper. Table 2 summarises the precise conditions on \(S_1\), \(S_2\), \(P_1\), \(P_2\) under which a given equilibrium type can occur for some values of v. Most of the conditions easily follow from the facts that the intervals described in Table 1 must be non-empty. The additional condition for the equilibrium \((S_2,P_1)\) (and \((S_1,P_2)\)) follows from (7). For equilibrium \((S_1,f(S_1))\) (and similarly for equilibrium \((f(P_1),P_1)\)) and \(\alpha <1\) the condition \(\frac{\alpha S_1^\alpha }{\bigl (1-(1-\alpha )^{1/\alpha }\bigr )^{\alpha +1}}\le 2\alpha (2S_1)^\alpha \) is always satisfied, since \((1-\alpha )^{1/\alpha }<\frac{1}{2}\); the condition \(\frac{\alpha S_1^\alpha }{\bigl (1-(1-\alpha )^{1/\alpha }\bigr )^{\alpha +1}}\le \frac{\alpha }{S_1}\bigl (S_1+P_2\bigr )^{\alpha +1}\) simplifies to \(\frac{S_1}{P_2}\le \frac{1-(1-\alpha )^{1/\alpha }}{(1-\alpha )^{1/\alpha }}\). For equilibrium \((S_1,P_1)\) the fact that the intervals in Table 1 are non-empty follows from (7) (and also by interchanging \(S_1\) and \(P_1\)). Further, these intervals have non-empty interior if and only if \(\frac{S_1+P_1}{S_1}\bigl ((S_1+P_1)^\alpha -P_1^\alpha \bigr )\le \frac{\alpha }{S_1}(S_1+P_1)^{\alpha +1}\) and \(\frac{S_1+P_1}{P_1}\bigl ((S_1+P_1)^\alpha -S_1^\alpha \bigr )\le \frac{\alpha }{P_1}(S_1+P_1)^{\alpha +1}\), which simplifies to the condition in Table 2. Finally, for equilibrium \((S_1,0)\) we can use (7) in case \(\alpha \ge 1\). From Table 2 we immediately obtain Table 3 and the necessary conditions shown in Table 4.

Table 3

Possible equilibria related to relative positions of \(S_1\), \(S_2\), \(P_1\), \(P_2\)

Relations between	Equilibrium
\(S_1\), \(S_2\), \(P_1\), \(P_2\)	1	2	3	4	5	6	7	8	9	10	11
\(S_1<S_2<P_1<P_2\)	•	⁎	⁎		⁎		•			•	•
\(S_1<P_1<S_2<P_2\)	•	⁎	⁎		•			•		•	•
\(S_1<P_1<P_2<S_2\)	•	⁎	⁎		•			•	•		•
\(P_1<P_2<S_1<S_2\)	•	⁎	⁎	⁎		•			•		•
\(P_1<S_1<P_2<S_2\)	•	⁎	⁎	•				•	•		•
\(P_1<S_1<S_2<P_2\)	•	⁎	⁎	•				•		•	•

‘•’ means that in this case the equilibrium is always possible for some value of v, ‘⁎’ means that for \(\alpha \ge 1\) the equilibrium is always possible for some value of v, while for \(\alpha <1\) there are certain additional conditions (to be found in Table 2) that need to be satisfied in order for the equilibrium to be present

Table 4

Necessary conditions

#	Equilibrium	Necessary condition
3	\((S_1,P_1)\)	\(v\le \frac{\alpha }{\max \{S_1,P_1\}}(S_1+P_1)^{\alpha +1}\le 2\alpha (S_1+P_1)^\alpha \)
4	\((S_1,f(S_1))\)	\(v\le \frac{\alpha }{S_1}\bigl (S_1+\min \{S_1,P_2\}\bigr )^{\alpha +1}\le 2\alpha (S_1+P_2)^\alpha \)
5	\((f(P_1),P_1)\)	\(v\le \frac{\alpha }{P_1}\bigl (\min \{S_2,P_1\}+P_1\bigr )^{\alpha +1}\le 2\alpha (S_2+P_1)^\alpha \)
8	\(\Bigl (\frac{1}{2}\bigl (\frac{v}{2\alpha }\bigr )^{\frac{1}{\alpha }},\frac{1}{2}\bigl (\frac{v}{2\alpha }\bigr )^{\frac{1}{\alpha }}\Bigr )\)	\(2\alpha (2\max \{S_1,P_1\})^\alpha \le v\le 2\alpha (2\min \{S_2,P_2\})^\alpha \)
9	\((f(P_2),P_2)\)	\(v\ge \frac{\alpha }{P_2}\bigl (\max \{S_1,P_2\}+P_2\bigr )^{\alpha +1}\ge 2\alpha (S_1+P_2)^{\alpha +1}\)
10	\((S_2,f(S_2))\)	\(v\ge \frac{\alpha }{S_2}\bigl (S_2+\max \{S_2,P_1\}\bigr )^{\alpha +1}\ge 2\alpha (S_2+P_1)^{\alpha +1}\)
11	\((S_2,P_2)\)	\(v\ge \frac{\alpha }{\min \{S_2,P_2\}}(S_2+P_2)^{\alpha +1}\ge 2\alpha (S_2+P_2)^\alpha \)

Now by comparing the conditions in various tables we can see why none of the equilibria when both players invest non-zero in the fights (equilibria numbered 3–11) ever occur together. This is summarised in Table 5.

Table 5

Reasons why intersection cannot occur

	4	5	6	7	8	9	10	11
3	Table 1	Table 1	Table 1	Table 1	Table 4	Table 4	Table 4	Table 4
4		Table 2	Table 1	Table 2	Table 1	Table 4	Table 1	Table 4
5			Table 2	Table 1	Table 1	Table 1	Table 4	Table 4
6				Table 2	Table 2	Table 1	Table 2	Table 1
7					Table 2	Table 2	Table 1	Table 1
8						Table 1	Table 1	Table 4
9							Table 2	Table 1
10								Table 1

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Authors and Affiliations

Mark Broom
- 1
Michal Johanis
- 2
Jan Rychtář
- 3
Email author

1.Department of MathematicsCity, University of LondonLondonUK
2.Department of Mathematical AnalysisCharles UniversityPrague 8Czech Republic
3.Department of Mathematics and StatisticsThe University of North Carolina at GreensboroGreensboroUSA

The effect of fight cost structure on fighting behaviour involving simultaneous decisions and variable investment levels

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Keywords

Mathematics Subject Classification

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Acknowledgements

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