Discrete and Continuous Distributed Delays in Replicator Dynamics

Abstract

In this paper, we study evolutionary games and we examine the stability of the evolutionarily stable strategy (ESS) in the continuous-time replicator dynamics with distributed time delays. In many examples, the interactions between individuals take place instantaneously, but their impacts are not immediate and take a certain amount of time which is usually random. In this paper, we study the consequences of distributed delays on the stability of the replicator dynamics. The main results are that (i) under the exponential delay distribution, the ESS is asymptotically stable for any value of the rate parameter; (ii) the necessary and sufficient conditions for the asymptotic stability of the ESS under uniform and Erlang distributions; and (iii) for the discrete distributed delays, we derive a necessary and sufficient delay-independent stability conditions. We illustrate our results with numerical examples from the Hawk–Dove game.

Appendices

Appendix A: Proof of Theorem 1

We suppose $\tau _\mathrm{max}<\frac{\pi ^2}{2D}$. If $\lambda$ is a real solution of (5), then it is clear that $\lambda$ cannot be positive. Let $\lambda =u+iv$ be a complex root of (5) with $u > 0$ and $v>0$ (without loss of generality we assume that $v>0$ since if $u+iv$ a solution of (5) then $u-iv$ is also a solution). We aim to find a contradiction, and hence we prove that no root of (5) with a positive real part can exist. We have:

$$\begin{aligned} v= & {} \frac{D}{v \tau _\mathrm{max}}\int _{0}^{\tau _\mathrm{max}v}{e^{-\alpha \frac{u}{v}}\mathrm{sin}(\alpha )\hbox {d}\alpha }. \end{aligned}$$

Let $\tau _\mathrm{max} v=2k\pi +\gamma$ with $k \ge 0$ and $0\le \gamma <2\pi$. We make this change of variable to explore the periodicity of sinus and cosinus functions. Hence, we get

$$\begin{aligned} v^2= & {} \frac{D}{\tau _\mathrm{max}}\int _{0}^{2k\pi +\gamma }{e^{-\alpha \frac{u}{v}}\mathrm{sin}(\alpha )\hbox {d}\alpha }\\\le & {} \frac{D (k+1)}{\tau _\mathrm{max}}\int _{0}^{\pi }{\mathrm{sin}(\alpha )\hbox {d}\alpha }=\frac{2(k+1)D}{\tau _\mathrm{max}}. \end{aligned}$$

Thus, we have $v^2 \le \frac{2(k+1)D}{\tau _\mathrm{max}}$; and since $\tau _\mathrm{max}<\frac{\pi ^2}{2D}$ we get $v \tau _\mathrm{max}< (k+1)^{\frac{1}{2}}\pi$. But $v \tau _\mathrm{max}=2k\pi +\gamma$ with $0 \le \gamma < 2\pi$, which finally yields $k=0$ and $v\tau _\mathrm{max}<\pi$.

On the other hand, we have $u=-\frac{D}{\tau _\mathrm{max}v}\int _{0}^{\tau _\mathrm{max}v}{e^{-\alpha \frac{u}{v}}\mathrm{cos}(\alpha )\hbox {d}\alpha }$. Let us study the sign of the right-hand side of this equation.

If $\tau _\mathrm{max}v \le \frac{\pi }{2}$ then we obtain $u<0$, this a contradiction.

If $\tau _\mathrm{max}v>\frac{\pi }{2}$, then we have

$$\begin{aligned} \int _{0}^{\tau _\mathrm{max}v}{e^{-\alpha \frac{u}{v}}\mathrm{cos}(\alpha )\hbox {d}\alpha }= \int _{0}^{\frac{\pi }{2}}{e^{-\alpha \frac{u}{v}}\mathrm{cos}(\alpha )\hbox {d}\alpha }+ \int _{\frac{\pi }{2}}^{\tau _\mathrm{max}v}{e^{-\alpha \frac{u}{v}}\mathrm{cos}(\alpha )\hbox {d}\alpha },\\ \ge e^{-\frac{\pi u}{2v}}\int _{0}^{\frac{\pi }{2}}{\mathrm{cos}(\alpha )\hbox {d}\alpha }+ e^{-\frac{\pi u}{2v}}\int _{\frac{\pi }{2}}^{\tau _\mathrm{max}v}{\mathrm{cos}(\alpha )\hbox {d}\alpha }>0. \end{aligned}$$

We obtain $u<0$ which is a contradiction with the initial assumption of $u > 0$. This proves the sufficient condition.

Now let us prove the necessary condition. A switch from stability to instability or the inverse corresponds to the existence of a root with a zero real part. We have to prove that when $\tau _\mathrm{max}=\frac{\pi ^2}{2D}$, the stability is lost and cannot be regained. Let $\lambda$ a solution of (5), and let $\lambda ^*=iw_0$ a pure imaginary solution of (5) and let $\tau _\mathrm{max}^*$ the value of $\tau _\mathrm{max}$ when $\lambda =\lambda ^*$. We will show that as $\tau _\mathrm{max}$ increases, the roots of the characteristic equation can cross the imaginary axis only from the left to the right, and therefore once the stability is lost, it cannot be regained. A stability switch occurs when $\int _{0}^{v\tau _\mathrm{max}^*}{\mathrm{cos}(\alpha )\hbox {d}\alpha }=0$ and $\int _{0}^{w_0\tau _\mathrm{max}^*}{\mathrm{sin}(\alpha )\hbox {d}\alpha } \not =0$ since zero is not a root of (5); which yields $w_0\tau _\mathrm{max}^*=(2k+1) \pi$. When $\tau _\mathrm{max}=\frac{\pi ^2}{2D}$ we check that the above two conditions are satisfied, and hence this value of $\tau _\mathrm{max}$ corresponds to a pure imaginary root. Moreover, at $\lambda =iv$ we have:

$$\begin{aligned} { Re}\, \left( \frac{\hbox {d}\lambda }{\hbox {d}\tau _\mathrm{max}}\right) _{\tau _\mathrm{max}=\tau _\mathrm{max}^*}=-Dw_0^2\mathrm{cos}(w_0\tau _\mathrm{max}^*) >0. \end{aligned}$$

The derivative of $\lambda (\tau _\mathrm{max})$ at $\tau _\mathrm{max}^*$ is positive, which means that roots can cross the imaginary axis only from the left to the right. Since the solution of (3) is asymptotically stable when $\tau _\mathrm{max}<\frac{\pi ^2}{2D}$, then the stability is lost at $\tau _\mathrm{max}=\frac{\pi ^2}{2D}$ and cannot be regained (because there is no possibility that all roots have negative real parts when further increasing the value of the delay). This proves the necessary condition.

Appendix B: Proof of Theorem 2

We shall prove that all roots of (6) have negative real parts for any $\beta >0$. Let $\alpha$ be a real solution of (6), it is clear that $\alpha$ cannot be positive. Let $\lambda =u+iv$ be a complex solution of (6) with $v>0$ (without loss of generality, we assume that $v>0$, since if $u+iv$ a solution of the characteristic equation, then $u-iv$ is also a solution). We aim to prove that no root with a positive real part can exist. Separating real and imaginary parts in (6), we get:

$$\begin{aligned} u+\beta D \int _{0}^{\infty }{e^{-(\beta +u)\tau } \mathrm{cos} (v\tau )\hbox {d}\tau }= & {} 0, \end{aligned}$$

(19)

$$\begin{aligned} v-\beta D \int _{0}^{\infty }{e^{-(\beta +u)\tau } \mathrm{sin} (\tau v)\hbox {d}\tau }= & {} 0. \end{aligned}$$

(20)

Furthermore, we have:

$$\begin{aligned} \int _{0}^{\infty }{e^{-(\beta +u)\tau } \mathrm{sin}(\tau v)\hbox {d}\tau }= & {} \frac{1}{v} \int _{0}^{\infty }{e^{-(\beta +u)\frac{z}{v}} \mathrm{sin}(z)dz} \nonumber \\= & {} \frac{I}{\beta +u}, \end{aligned}$$

with $I=\int _{0}^{\infty }{e^{-(\beta +u)\frac{z}{v}}\mathrm{cos }(z)dz}$.

Taking into account the above equations, we can write (19) and (20) as follows:

$$\begin{aligned} u+\beta D \frac{I}{v}= & {} 0, \nonumber \\ v-\beta D \frac{I}{\beta +u}= & {} 0. \end{aligned}$$

The two above equations yield $u=\frac{-\beta }{2}<0$. Therefore, the real parts of the roots are always negative, which results in the asymptotic stability.

Appendix C: Proof of Theorem 3

Let $\lambda =u+iv$ be a solution of the characteristic equation with $u>0$. We suppose without loss of generality that $v>0$ (recall that if $u+iv$ is a solution of the characteristic equation then $u-iv$ is also a solution). We aim to prove that if $u>0$, then $\beta <\beta _c$. By substituting $\lambda$ with $u+iv$ and separating real and imaginary parts in the characteristic equation, we get :

$$\begin{aligned} u+[(\beta +u)^2+v^2]^{-\frac{k}{2}} D \beta ^k\mathrm{cos}(k\theta )= & {} 0,\\ v-[(\beta +u)^2+v^2]^{-\frac{k}{2}}D \beta ^k\mathrm{sin}(k\theta )= & {} 0, \end{aligned}$$

with $\mathrm{cos}(\theta )=\frac{\beta +u}{[(\beta +u)^2+v^2]^\frac{1}{2}}$ and $\mathrm{sin}(\theta )=\frac{v}{[(\beta +u)^2+v^2]^\frac{1}{2}}$ (with $0 \le \theta < 2\pi$). The above system can be written:

$$\begin{aligned} u= & {} \frac{-D \beta ^k\mathrm{cos}(k\theta )}{[(\beta +u)^2+v^2]^{\frac{k}{2}} },\\ v= & {} \frac{D \beta ^k\mathrm{sin}(k\theta )}{[(\beta +u)^2+v^2]^{\frac{k}{2}}}. \end{aligned}$$

From the equations above, we get:

$$\begin{aligned}&\mathrm{cos}(k\theta )<0 \Rightarrow \frac{\pi }{2k}<\theta , \nonumber \\&v^2<D^2\frac{\beta ^{2k}}{[(\beta +u)^2+v^2]^k}<D^2\frac{\beta ^{2k}}{[\beta ^2+v^2]^k}; \end{aligned}$$

(21)

By virtue of (21), we get $\mathrm{cos}(\theta )<\mathrm{cos}(\frac{\pi }{2k})$. In addition, we check that $\frac{\beta }{[\beta ^2+v^2]^{\frac{1}{2}}}<\frac{\beta +u}{[(\beta +u)^2+v^2]^\frac{1}{2}}$ (recall that u is positive); and then we have $\frac{\beta }{[\beta ^2+v^2]^{\frac{1}{2}}}<\mathrm {cos}(\theta )<\mathrm{cos}(\frac{\pi }{2k})$. From this inequality, we obtain:

$$\begin{aligned} \beta ^2\frac{\mathrm{sin}^2(\frac{\pi }{2k})}{ \mathrm{cos}^2(\frac{\pi }{2k})}<v^2. \end{aligned}$$

(22)

On the other hand, we have $v^2<D^2\frac{{\beta }^{2k}}{[\beta ^2+v^2]^k}$ and $\frac{\beta ^{2k}}{[\beta ^2+v^2]^k}<\mathrm{cos}^{2k}(\frac{\pi }{2k})$, which yields:

$$\begin{aligned} v^2<D^2 \mathrm{cos}^{2k}(\frac{\pi }{2k}). \end{aligned}$$

(23)

Using (22) and (23), we get $\beta <\beta _c=D \frac{\mathrm{cos}^{k+1}(\frac{\pi }{2k})}{\mathrm{sin}(\frac{\pi }{2k})}$. Therefore, we conclude that there is no root with a positive real part when $\beta >\beta _c$, and the local asymptotic stability follows.

Now, let us prove the necessary condition. By implicit differentiation of the characteristic equation, we have:

$$\begin{aligned} \mathcal{{R}}e \Big (\frac{\hbox {d}\lambda }{\hbox {d}\beta } \Big )_{\beta ={\beta _c}}<0. \end{aligned}$$

Therefore, as $\beta$ decreases, the roots of characteristic equation cross the imaginary axis only from the left to the right and once the stability is lost, it cannot be regained again. The stability is persistently lost at $\beta ={\beta _c}$.

Appendix D: Proof of Theorem 4

The characteristic equation is given by:

$$\begin{aligned} \lambda -\gamma (p_1(a-b)-c+d)-\gamma (1-p_1)(a-b)e^{-\lambda \tau }=0. \end{aligned}$$

Let $\lambda =u+iv$ be a root of the characteristic equation with $v>0$ (without loss of generality). We aim to prove that when $u>0$, $\tau >\tau _c$. By separating real and imaginary parts in the characteristic equation, we get:

$$\begin{aligned} u-A-B\text{ exp }(-u\tau )\text{ cos }(v\tau )= & {} 0, \end{aligned}$$

(24)

$$\begin{aligned} v+B\text{ exp }(-u\tau )\text{ sin }(v\tau )= & {} 0 , \end{aligned}$$

(25)

where $A=\gamma (p_1(a-b)-c+d)$ and $B=\gamma (1-p_1)(a-b)$. We have the relation $A>B$ and $A+B<0$. First, we study the case when $A<0$ and $B<0$. From (24) and (25), we have:

$$\begin{aligned} \text{ cos }(v\tau )\le & {} -\frac{A}{B} \quad \text{(recall } \text{ that } u>0), \end{aligned}$$

(26)

$$\begin{aligned} v^2\le & {} B^2-A^2. \end{aligned}$$

(27)

The system above yields $\tau >\tau _c$. Therefore, we conclude that when $\tau <\tau _c$, the real parts of the roots of the characteristic equation are negative.