One problem, too many solutions: How costly is honest signalling of need? - PubMed (original) (raw)
One problem, too many solutions: How costly is honest signalling of need?
Szabolcs Számadó et al. PLoS One. 2019.
Abstract
The "cost of begging" is a prominent prediction of costly signalling theory, suggesting that offspring begging has to be costly in order to be honest. Seminal signalling models predict that there is a unique equilibrium cost function for the offspring that results in honest signalling and this cost function must be proportional to parent's fitness loss. This prediction is only valid if signal cost and offspring condition is assumed to be independent. Here we generalize these models by allowing signal cost to depend on offspring condition. We demonstrate in the generalized model that any signal cost proportional to the fitness gain of the offspring also results in honest signalling. Moreover, we show that any linear combination of the two cost functions (one proportional to parent's fitness loss, as in previous models, the other to offspring's fitness gain) also leads to honest signalling in equilibrium, yielding infinitely many solutions. Furthermore, we demonstrate that there exist linear combinations such that the equilibrium cost of signals is negative and the signal is honest. Our results show that costly signalling theory cannot predict a unique equilibrium cost in signalling games of parent-offspring conflicts if signal cost depends on offspring condition. It follows, contrary to previous claims, that the existence of parent-offspring conflict does not imply costly equilibrium signals. As an important consequence, it is meaningless to measure the "cost of begging" as long as the dependence of signal cost on offspring condition is unknown. Any measured equilibrium cost in case of condition-dependent signal cost has to be compared both to the parent's fitness loss and to the offspring's fitness gain in order to provide meaningful interpretation.
Conflict of interest statement
The authors have declared that no competing interests exist.
Figures
Fig 1. Inclusive fitness functions and optima without signalling cost.
(A) G = ½; (B) G = 0.08 (as in [7]). Inclusive fitness functions parameterized by z are shown as yellow curve for parent’s (function u according to Eq 2 without signal cost f(x)) and (blue curve for offspring (function v according to Eq 1 without signal cost f(x)). The x coordinate value of parent’s curve is the parent’s own fitness contribution g(c, z), the y coordinate value is the fitness contribution the offspring (γ h(c, z)); similarly, the x value of the offspring’s curve is the parent’s contribution (ψ g(z)), the y value is the offspring’s own fitness h(c, z). The actual inclusive fitness value is the sum of the appropriate coordinate values, both for parent and offspring. Parameters are Z = 2, γ = 1/2, ψ = ½, U = 1, c = 3. Yellow and blue stars indicate parent’s and offspring’s optima, respectively. Dashed lines are the calculated derivative tangents that touch optima at 45°, indicating maximum fitness. The optimum z value for parent and offspring are not identical: the yellow dot indicates what the parent’s fitness is at the offspring’s optimum z; blue dot is the offspring’s fitness in case of parent’s optimum z.
Fig 2. Inclusive fitness depending on offspring condition c and parental investment z.
(A) Parental fitness (function u according to Eq 2 without signal cost f(x)). (B) Offspring fitness (function v according to Eq 1 without signal cost f(x)). Red line connects the equilibrium z values where z˜=ln(γUcG)/c holds. Parameters are Z = 10, G = 0.08, U = 1, γ = ½, ψ = ½.
Fig 3. Inclusive fitness functions and optima with signalling cost.
Inclusive fitness functions, parameterized by z, are represented by the yellow curve (for the parent, fitness function u according to Eq 2) and by the blue curve (for the offspring, fitness function v according to Eq 1). The x coordinate value of parent’s curve is the parent’s own fitness contribution g(c, z), the y coordinate value is the fitness contribution of all future offspring (γ h(c, z)); similarly, the x value of the offspring’s curve is the parent’s contribution (minus cost) (ψ g(z) – L(α, c, z)), the y value is the offspring’s own fitness h(c, z). The actual inclusive fitness value is the sum of the appropriate coordinate values, both for parent and offspring. Parameters are Z = 2, γ = ½, ψ = ½, U = 1, G = ½, c = 3. Yellow and blue stars indicate parent’s and offspring’s fitness optima. Dashed lines are the calculated derivative tangents that touch optima at 45°, indicating maximum fitness. The optimum z value for parent and offspring are always identical, regardless of α and β values. (A) Cost function _L_1 of Nöldeke and Samuelson [17] (Eq 12; α = 1). (B) Cost function _L_2 introduced in this paper (Eq 16; α = 0). (C) Linear combination of the above two cost functions _L_1 and _L_2 (α = ½) (Eq 18).
Fig 4. Signalling cost functions, depending on offspring condition c and parental transfer z.
(A) equilibrium signal cost (f(x*(c))) for the different cost functions (Eqs 13, 17 and 19). (B) Signal cost function _L_1 (Eq 12). (C) Linear combination ½ _L_1 + ½ _L_2 (Eq 18) (D) signal cost function _L_2 introduced in this paper (Eq 16). Red, green and orange curves show the signal cost (f(x*(c))) along the equilibrium path (which describes the equilibrium transfer function z˜(c) for parent as a function of c); panel A shows these curves projected to the _c_-f(c) plane. (E) Partial derivatives of the signal cost functions _L_1, _L_2 and their linear combination, with respect to z along the equilibrium path as a function of c. (F) Partial derivative of signal cost function _L_1 (Eq 12) with respect to z. (G) Partial derivative of the linear combination ½ _L_1 + ½ _L_2 (Eq 18) (H) Partial derivative of signal cost function _L_2 introduced in this paper (Eq 16) with respect to z. Red, green and orange curves show the partial derivatives of the respective signal cost functions along the equilibrium path with respect to z; panel E shows these curves projected to the _c_-f(c) plane. Parameters are Z = 10, G = 0.08, U = 1, γ = ½, ψ = ½.
Fig 5. Negative equilibrium cost.
(B) Cost of signalling L depending on offspring condition c and amount of transferred resource z (Eq 18). Red-green curve along the surface (and in inset) indicates equilibrium cost function (f(x*(c))) at equilibrium z˜ (Eq 19), where red sections indicate negative cost values. (A) shows this curve projected to the _c_-f(c) plane. Parameters are: {Z=4,G=0.1,U=1,γ=12,ψ=12,cmax=5,α=5}.
References
- Trivers R. L. 1974. Parent-offspring conflict. Integrative and Comparative Biology 14, 249–264.
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