3.1. Formal derivation of mean-field limit

We begin by considering the model without noise (i.e. $\sigma = 0$ ) and deriving a PDE satisfied by the empirical density, defined by

(6) \begin{equation} \rho (t,a,x) = \frac {1}{N} \sum _{i=1}^N \delta _{a_i(t)}(a) \, \delta _{x_i(t)}(x) , \end{equation}

where $\delta _{z}$ denotes a Dirac delta centred at $z$ . Let $\zeta (a,x)$ be an infinitely differentiable, compactly supported test function. We formally compute

\begin{align*} & \int _U \int _0^A \partial _t\rho (t,a,x) \,\zeta (a,x) \, \text{d}a \, \text{d}x \\ & \quad = \frac{\text{d}}{\text{d}t} \int _U \int _0^A \frac {1}{N} \sum _{i=1}^N \delta _{a_i(t)}(a) \, \delta _{x_i(t)}(x) \, \zeta (a,x) \, \text{d}a \, \text{d}x \\ &\quad = \frac{\text{d}}{\text{d}t} \frac {1}{N} \sum _{i=1}^N \zeta \big (a_i(t), x_i(t)\big ) \\ &\quad = \frac {1}{N} \sum _{i=1}^N \partial _a \zeta \big (a_i(t), x_i(t)\big ) \,\tau + \frac {1}{N} \sum _{i=1}^N \partial _x \zeta \big (a_i(t), x_i(t)\big ) \left( \frac {1}{N} \sum _{j=1}^N M(a_i,a_j) \, \varphi (x_j - x_i) \right) \\ &\quad = \int _U \int _0^A \frac {1}{N} \sum _{i=1}^N \delta _{a_i(t)}(a) \, \delta _{x_i(t)}(x) \left( \partial _a \zeta (a, x) \,\tau + \partial _x \zeta (a, x) \, \frac {1}{N} \sum _{j=1}^N M(a,a_j) \, \varphi (x_j - x) \right) \, \text{d}a \, \text{d}x \\ &\quad = \int _U \int _0^A \rho (t,a,x) \, \partial _a \zeta (a, x) \,\tau + \partial _x \zeta (a, x) \, \rho (t,a,x) \left( \int _U \int _0^A M(a,b) \, \varphi (y - x) \, \rho (t,b,y) \, \text{d}b \, \text{d}y \, \right) \, \text{d}a \, \text{d}x \\ &\quad = - \int _U \int _0^A \zeta (a, x) \, \partial _a\rho (t,a,x) \,\tau\\[5pt] & \qquad + \zeta (a, x) \, \partial _x \left( \rho (t,a,x) \left( \int _U \int _0^A M(a,b) \, \varphi (y - x) \, \rho (t,b,y) \, \text{d}b \, \text{d}y \, \right) \right) \, \text{d}a \, \text{d}x \end{align*}

Note that in the final line, the integration by parts produces no boundary terms since $\zeta$ is compactly supported. This gives that $\rho$ is a weak solution to

(7) \begin{equation} \partial _t\rho (t,a,x) + \tau \partial _a\rho (t,a,x) + \partial _x \left( \rho (t,a,x) \left( \int _U \int _0^A M(a,b) \, \varphi (y - x) \, \rho (t,b,y) \, \text{d}b \, \text{d}y \, \right) \right) = 0 \,. \end{equation}

When considering the model with noise, the derivation is more complex as the empirical density no longer satisfies the limiting PDE. Indeed, the stochastic effects only disappear in the large-population limit [Reference Jabin and Wang30]. In this case, we note that the structure of (1) is similar to that of the second-order Cucker-Smale model for bird flocking [Reference Cucker and Smale10] and the general second-order model discussed in [Reference Jabin and Wang30], in which alignment and noise are only present in the evolution of velocities (here opinions). The derivation of the mean-field limit in this setting can be found in [Reference Graham, Kurtz and Méléard23, Reference Sznitman51]. While we do not show this rigorously here, the similarity in structure implies that the corresponding mean-field limit for (1) is given by

(8) \begin{equation} \partial _t\rho + \tau \partial _a\rho + \partial _x \left( \rho (t,a,x) \left( \int _U \int _0^A M(a,b) \, \varphi (y - x) \, \rho (t,b,y) \, \text{d}b \, \text{d}y \, \right) \right) - \, \frac {\sigma ^2}{2} \partial _{xx} \rho = 0 \,. \end{equation}

Defining the flux,

(9) \begin{align} F[\rho ](t,a,x) = \rho (t,a,x) \left( \int _U \int _0^A M(a,b) \, \varphi (y - x) \, \rho (t,b,y) \, \text{d}b \, \text{d}y - \, \frac {\sigma ^2}{2} \partial _{x}\log (\rho (t,a,x)) \right) , \end{align}

then the PDE can be written as

\begin{align*} \partial _t \rho + \tau \partial _a \rho + \partial _x F[\rho ] = 0 \,. \end{align*}

Note that the flux also incorporates the diffusion term.

We also need to specify the initial and boundary conditions. We assume that the initial condition is a given probability distribution $\rho _0(a,x)$ . As we imposed reflecting boundary conditions in the SDE model, we impose no-flux boundary conditions on the boundary of $U$ (here $x=\pm 1$ ) [Reference Erban and Chapman15, Reference Goddard, Gooding, Short and Pavliotis22]. It is also necessary to specify the opinion distribution at age-zero. As with the microscopic system, we assume that this follows a given distribution $\mu$ . In order to maintain a constant population size, we therefore set

\begin{align*} \rho (t,0,x) = \mu (x) \, \int _U \rho (t,1,y) \, \text{d}y , \end{align*}

as the integral

\begin{align*} \int _U \rho (t,1,y) \, \text{d}y , \end{align*}

describes the total volume of individuals whose opinions are reset at time $t$ .

The complete initial-boundary value problem can be written

(10a) \begin{align} \partial _t \rho (t,a,x) + \tau \partial _a \rho (t,a,x) + \partial _x F[\rho ](t,a,x) &= 0\\[-10pt]\nonumber \end{align}

(10b) \begin{align} F[\rho ](t,a,-1) = F[\rho ](t,a,1) &= 0\\[-10pt]\nonumber \end{align}

(10c) \begin{align} \rho (t,0,x) &= \mu (x) \, \int _U \rho (t,1,y) \, \text{d}y\\[-10pt]\nonumber \end{align}

(10d) \begin{align} \rho (0,a,x) &= \rho _0(a,x) \end{align}

Having derived this PDE problem, we next look at several example solutions before moving on to our analysis of its steady states.

Note that this age-structured PDE (10) can also be compared against the standard mean-field limit for the model without age structure:

(11a) \begin{align} \partial _t \rho (t,x) + \partial _x \left( \rho (t,x) \left( \int _U \varphi (y - x) \, \rho (t,y) \, \text{d}y \, \right) \right) - \, \frac {\sigma ^2}{2} \partial _{xx} \rho (t,x) &= 0\\[-10pt]\nonumber \end{align}

(11b) \begin{align} \rho (t,1) \left( \int _U \varphi (y - 1) \, \rho (t,y) \, \text{d}y \, \right) - \, \frac {\sigma ^2}{2} \partial _{x} \rho (t,1) &= 0\\[-10pt]\nonumber \end{align}

(11c) \begin{align} \rho (t,-1) \left( \int _U \varphi (y + 1) \, \rho (t,y) \, \text{d}y \, \right) - \, \frac {\sigma ^2}{2} \partial _{x} \rho (t,-1) &= 0\\[-10pt]\nonumber \end{align}

(11d) \begin{align} \rho (0,x) &= \rho _0(x) \,. \end{align}

3.2. Numerical simulations of the mean-field PDE

In this section, we show numerical solutions of (10), some of which mirror the examples in Section 2. A description of the numerical scheme used can be found in Appendix B.1.

In each of the examples shown in Figure 2, we take both the initial distribution $\rho _0(a,x)$ and the age-zero distribution $\mu$ to be uniform (with the exception of Figure 2f). We again use the mollified bounded confidence interaction function (2). By varying the confidence bounds, ageing rate and the strength of the noise/diffusion, we observe different clustering behaviours. In each figure, we show the evolution over time of the total opinion density,

(12) \begin{align} P(t,x) = \int \rho (t,a,x) \, \text{d}a, \end{align}

for $x\in U,\, t \geq 0$ . This allows a more direct comparison with the behaviours observed in the microscopic model.

Figure 2.

Example numerical solutions to (10). Colour indicates the total opinion density $P(t,x)$ (12). The colour map is shifted to make areas of low density more clearly visible. All examples use an interaction function $\phi$ given by (2), $M \equiv 1$ and a uniform $\mu$ and $\rho _0$ , with the exception of Figure 2f which uses (4) for $\rho _0$ . The values of parameters $r_1, r_2, \tau$ and $\sigma$ are varied to demonstrate different behaviours, with the final time $T$ changed accordingly. All solutions use the numerical scheme described in Appendix B.1 with $J_x = J_a = 500$ and $\Delta t = 0.001$ .

Figures 2a and 2b show examples in which the population quickly moves towards one and two stable clusters, respectively, with Figure 2b mirroring the behaviour observed in the microscopic system in Figure 1b. Here, the ageing rate $\tau$ and level of noise $\sigma$ have a relatively minor effect compared to the strong clustering. Reducing either $\tau$ or $\sigma$ leads to more strongly peaked clusters, as individuals either have longer lifespans in which to concentrate their opinions or diffuse less away from the centre of the cluster. These effects can be seen by comparing Figures 2a and 2b, in the former $\tau$ and $\sigma$ are relatively larger and the cluster(s) formed are therefore broader. In Figure 2c, we show an intermediate point between the formation of one and two clear clusters. A strong central cluster forms, initially bringing the whole population to consensus. However, as new individuals enter the population, those with more extreme opinions are not brought into this consensus. This leads to the formation of two smaller, less concentrated clusters at the extremes.

In Figure 2d, we consider a smaller confidence bound. Initially, the population remains near the uniform distribution due to the relatively strong diffusion, except at the boundary of the opinion space. As individuals at the boundary can only interact with those with less extreme opinions, small clusters form near the boundary, causing a subsequent decrease in density on the opposite side of the cluster. This, in turn, prompts the formation of another cluster closer to the centre, repeating until these clusters reach the centre of the opinion space. We also observe that by time $t = 400$ this central cluster cannot be maintained, and all individuals in this cluster either die or join the clusters on either side. As in Figure 2b, the population reaches a steady state with almost evenly spaced, although not evenly sized, clusters.

Figures 2e and 2f show instances of periodic behaviour, similar to that seen in the microscopic model in Figure 1d. In Figure 2e, as in 2c, a strong central cluster quickly forms, leaving a vacuum at the extreme opinions in which new, smaller clusters form. In this case, these eventually combine with the central cluster before the cycle repeats again. This effect is only possible due to the influx of new individuals to the population, as without it the initial consensus would simply persist. The parameter range for which this behaviour occurs is relatively small, as it lies on the boundary between the population forming into two clusters (as in Figure 2b) and the population forming one large central cluster and two smaller clusters that do not merge (as in Figure 2c).

In Figure 2f, we begin with a non-uniform initial condition given by (4), which has two peaks centred at $0$ and $-0.8$ . Due to the uniform influx of new individuals, these peaks move to become more symmetric. As in Figure 2e, these clusters periodically combine and reform. However, in this case, a lack of symmetry persists, meaning that after the two clusters combine, only one new extreme cluster forms (rather than one forming near each boundary). This leads to a periodic shifting between positive and negative clusters that form a brief off-centre consensus that cannot be maintained. This directly mirrors the behaviour observed on the individual level in Figure 1d.

These examples demonstrate the rich variety of complex behaviours that are made possible by the inclusion of a continuous age-structure. In the following Section 4, we move to studying the possible stationary states of this model.

4.1. Existence of steady states

In this section, we make the following assumptions:

Assumption 2A means that the initial ( $t=0$ ) age profile is uniform. As will be seen in Section 5.1, the age profile of the population satisfies a transport equation, so Assumption 2A ensures that the age profile remains uniform at all times. Assumption 2B ensures that all individuals view the population-level opinion density in the same way, regardless of their own age. This assumption will be crucial in performing the analysis below. As discussed in Section 5.6, even with these two assumptions, the model can still effectively capture realistic ageing effects and non-uniform population-level age distributions. Note that Assumption 2C can be achieved by appropriately rescaling $A$ , $\tau$ and $\sigma$ and is made to reduce the number of parameters.

As the age-interaction kernel $M$ is Lipschitz continuous on a finite domain $U$ , it is also bounded and we denote the maximum value of $M$ by

\begin{align*} M_{\max } = \max \limits _{0\leq b \leq A} M(b) \,. \end{align*}

We can now rewrite (13a) as

\begin{align*} \tau \partial _a\rho (a,x) + \partial _x \left( \rho (a,x) \left( \int _U \varphi (y - x) \bigg ( \int _0^A M(b) \, \rho (b,y) \, \text{d}b \bigg ) \, \text{d}y \, \right) \right) - \, \frac {\sigma ^2}{2} \partial _{xx} \rho (a,x) &= 0 \,. \end{align*}

To show the existence of a steady state, we will use a fixed-point argument. To this end, assume temporarily that the value of the following integral, which is constant in $a$ , is known

\begin{align*} \lambda (y) = \int _0^A M(b) \, \rho (b,y) \, \text{d}b \,. \end{align*}

This integral describes the population-level opinion distribution with which individuals interact. From this, we define $\Lambda \, : \, U\rightarrow \mathbb{R}$ by

(16) \begin{align} \Lambda (x) \, :\!= \, \int _U \varphi (y - x) \, \lambda (y) \, \text{d}y , \end{align}

and let $\tilde {\rho }(a,x)$ be a solution to the initial-boundary value problem

(17a) \begin{align} \partial _a\rho (a,x) + \partial _x \Big ( \rho (a,x) \Lambda (x) \Big ) - \, \frac {\sigma ^2}{2} \partial _{xx} \rho (a,x) &= 0 , \end{align}

(17b) \begin{align} \rho (0,x) &= \mu (x) ,\\[-10pt]\nonumber \end{align}

(17c) \begin{align} \rho (a,1) \, \Lambda (1) - \, \frac {\sigma ^2}{2} \partial _{x} \rho (a,1) &= 0 ,\\[-10pt]\nonumber \end{align}

(17d) \begin{align} \rho (a,-1) \, \Lambda ({-}1) - \, \frac {\sigma ^2}{2} \partial _{x} \rho (a,-1) &= 0 \,. \end{align}

From this solution, define

(18) \begin{align} \tilde {\lambda }(y) &= \int _0^A M(b) \, \tilde {\rho }(b,y) \, \text{d}b , \end{align}

and note that if $\tilde {\lambda } = \lambda$ we have in fact recovered (13a). That is, there is a one-to-one correspondence between stationary states of (10) and fixed points of the mapping $\lambda \mapsto \tilde {\lambda }$ described above. In Appendix A, we rigorously show the existence of such stationary states through the following steps:

1. Step 1: We determine conditions for the existence of solutions to (17) in an appropriate Sobolev space, applying results from [Reference Ladyzhenskaia, Solonnikov and Ural’tseva34] to show such existence in PropositionA.2.

2. Step 2: Introduce the mapping $\lambda \mapsto \tilde {\lambda } \, =\!: \, \mathcal{F}(\lambda )$ and show that $\mathcal{F} \, : \, L^\infty (U)\rightarrow L^\infty (U)$ is well-defined. Specifically,

   \begin{align*} \mathcal{F}(\lambda )(x) = \int _0^A M(b) \rho (b,x) \, \text{d}b \,. \end{align*}

3. Step 3: Find an appropriate set $K\subset L^\infty (U)$ so that we may apply Schauder’s Fixed Point Theorem to $\mathcal{F} \, : \, K\rightarrow K$ . Using PropositionsA.3–A.6, we establish the existence of a constant $C$ such that the set

   \begin{align*} K = \{ \lambda \in L^\infty (U) \, : \, \lambda \geq 0 \text{ a.e.}, \| \lambda \|_{L^1(U)}=1, \, \| \lambda \|_{L^\infty (U)}\leq C \} \end{align*}
   satisfies the requirements.

This allows us to show the following theorem.

A natural question to ask next is whether such a steady state is unique. More specifically, when $\tau$ is large, individuals are quickly replaced and the opinion distribution cannot deviate significantly from the age-zero distribution $\mu$ , so we may expect uniqueness of the steady state. Indeed, the following Proposition4.1, proven in Appendix A, states that the mapping $\mathcal{F}$ is Lipschitz continuous on $K$ .

Proposition 4.1. Under the assumptions of Theorem 4.1, the mapping $\mathcal{F} \, : \, K\rightarrow K$ is Lipschitz continuous. Specifically there exists some constant $c$ depending on $U$ , $\phi$ , $A$ and $\tau$ but independent of $\lambda$ , such that for all $\lambda _1,\lambda _2 \in K$ ,

(19) \begin{align} \big \| \mathcal{F}(\lambda _1) - \mathcal{F}(\lambda _2) \big \|_{L^\infty (U)} \leq c \, \big (A^{1/2}+ A^{-1/4} \big )\, M_{\max } \, \big \|\lambda _1 - \lambda _2\big \|_{L^\infty (U)} \,. \end{align}

However, it is not possible to bound $c_8$ such that this Lipschitz constant is smaller than $1$ as $\tau \rightarrow \infty$ . Therefore, we cannot conclude that $\mathcal{F}$ is a contraction for sufficiently large $\tau$ (which would allow an application of the Banach Fixed-Point Theorem). Section 4.3 explores numerically an example in which steady states are and are not unique as $\tau$ varies.

4.2. Stationary age-zero density

We first describe a specific situation in which it is straightforward to identify a steady state. Assume that the age-zero distribution $\mu$ is a steady state of the classical mean-field PDE. That is,

(20) \begin{align} \partial _x \left( \mu (x) \left( \int _U \varphi (y - x) \, \mu (x) \, \text{d}y \, \right) \right) - \, \frac {\sigma ^2}{2} \partial _{xx} \mu (x) &= 0 , \end{align}

coupled with the appropriate no-flux boundary conditions. Define $\rho ^*(a,x) = \mu (x)$ for all $a \in [0,A]$ . Then we have,

\begin{align*} & \tau \partial _a\rho ^*(a,x) + \partial _x \left( \rho ^*(a,x) \left( \int _U \int _0^A M(a,b) \, \varphi (y - x) \, \rho ^*(b,y) \, \text{d}b \, \text{d}y \, \right) \right) - \, \frac {\sigma ^2}{2} \partial _{xx} \rho ^*(a,x) \\[3pt] &= \tau \partial _a\mu (x) + \partial _x \left( \mu (x) \left( \int _U \int _0^A M(a,b) \, \varphi (y - x) \, \mu (y) \, \text{d}b \, \text{d}y \, \right) \right) - \, \frac {\sigma ^2}{2} \partial _{xx} \mu (x) \\[3pt] &= \partial _x \left( \mu (x) \left( \int _U \, \varphi (y - x) \, \mu (y) \, \text{d}y \right) \left( \int _0^A M(a,b) \, \text{d}b \, \right) \right) - \, \frac {\sigma ^2}{2} \partial _{xx} \mu (x) \\[3pt] &= \partial _x \left( \mu (x) \left( \int _U \, \varphi (y - x) \, \mu (y) \, \text{d}y \right) \right) - \, \frac {\sigma ^2}{2} \partial _{xx} \mu (x) \\[3pt] &= 0 , \end{align*}

hence (13a) is satisfied. In addition

\begin{align*} \rho ^*(0,x) - \mu (x) \, \int _U \rho ^*(1,y) \, \text{d}y &= \mu (x) - \mu (x) \, \int _U \rho ^*(1,y) \, \text{d}y = \mu (x) - \mu (x) = 0\, \end{align*}

hence (13b) is satisfied. Finally, the no-flux boundary conditions for $\mu$ directly imply those for $\rho ^*$ , hence all conditions are satisfied for $\rho ^*$ to be a steady state. We now use this context to provide a specific example of non-uniqueness and demonstrate the impact of changing $\tau$ .

4.3. Non-uniqueness

As discussed at the end of Section 4.1, we cannot show that steady states are unique. That is, for given $\mu$ and $\phi$ , there may be multiple $\lambda \in K$ such that the corresponding solution $\rho \in W_2^{2,1}(U_A)$ of (17) also satisfies $\lambda = \mathcal{F}(\lambda )$ . In Figure 3 below, we give a specific example in the setting discussed in the previous Section 4.2, in which $\mu$ is stationary in the classical mean-field limit.

Consider an age-zero distribution $\mu ^{(2)}$ satisfying (20) in which the population is split into two equal clusters (as in e.g. [Reference Goddard, Gooding, Short and Pavliotis22]). Note that $\mu ^{(2)}$ is found numerically using a scheme that applies the opinion operator described in Appendix B.1. As in the previous Section 4.2, define $\rho ^*(a,x) = \mu ^{(2)}(x)$ for all $a\in [0,A]$ . As this $\rho ^*$ is known to be a steady state, we also have that $\mathcal{F}(\mu ^{(2)}) = \mu ^{(2)}$ .

We can also consider iteratively applying $\mathcal{F}$ from a different starting point, using a numerical approximation of $\mathcal{F}$ , described in Appendix B.2,. In particular, we begin from a solution $\mu ^{(1)}$ of (20) with only a single cluster. We find that we do not converge towards $\mu ^{(2)}$ but instead reach a different steady state in which the population is encouraged towards consensus at higher ages.

An example is shown in Figure 3, using a smoothed bounded confidence interaction function with $r_1 = 0.5, r_2 = 0.6$ , $\tau = 0.15$ , $\sigma = 0.05$ and $M\equiv 1$ . Heatmaps show the stationary distribution over $U_A$ , with age on the horizontal axis and opinion on the vertical axis. In the left panel, we begin at $\mu ^{(2)}$ and the population remains in these two stable clusters. However, in the right panel, we begin the fixed point iteration from $\mu ^{(1)}$ and observe a different stationary distribution. This shows that in certain cases, multiple stable behaviours could be observed on the macroscopic level.

Figure 3.

Showing two alternative steady states of (10) over the joint age-opinion space. In both cases the interaction function $\phi$ is a smooth bounded confidence (2) with $r_1 = 0.5, r_2 = 0.6$ , the diffusion coefficient $\sigma = 0.05$ , the age-interaction kernel $M \equiv 1$ and the age-zero distribution $\mu = \mu ^{(2)}$ . The first corresponds to a fixed point of the mapping (A6) with $\lambda _0 = \mu ^{(2)}$ , while the second corresponds to a fixed point in which $\lambda _0= \mu ^{(1)}$ has a single cluster.

We next investigate the effect of increasing $\tau$ . For each value of $\tau$ , we again begin the fixed point iteration at $\mu ^{(1)}$ . The final distribution reached is shown for various $\tau$ in Figure 4. For small values below $\tau = 0.23$ (including the example shown in the right of Figure 3), $\lambda$ has a single peak, indicating that the population merges into a single cluster as individuals age. As $\tau$ is increased, the age at which individuals merge increases, causing $\lambda$ to spread out. Eventually, this merging occurs extremely close to age $a=1$ and $\lambda$ has two peaks. Finally, for $\tau$ above approximately $0.323$ , the stationary state in which the clusters merge no longer exists, and $\lambda = \mu ^{(2)}$ is the unique stationary state.

Figure 4.

The fixed point $\lambda$ of the mapping $\mathcal{F}$ (A6), calculated numerically for various values of $\tau$ . The interaction function $\phi$ is a smooth bounded confidence (2) with $r_1 = 0.5, r_2 = 0.6$ , the diffusion coefficient $\sigma = 0.05$ , the age-interaction kernel $M \equiv 1$ and the age-zero distribution $\mu = \mu ^{(2)}$ . As $\tau$ is increased, there is a transition from a density with a single cluster to a density with two nearby peaks, then further to $\lambda = \mu ^{(2)}$ .

This example raises the interesting question of what clustering behaviours, merges and emergences are possible as the opinion profile changes (in $a$ ) across a steady state? This forms a part of a broader question about the type of patterns it is possible to observe in the joint age-opinion distribution at a steady state, and which of these are stable.

It is also natural to consider the stability of these stationary distributions. However, performing even a linear stability analysis of this system is not straightforward for several reasons: the stationary distributions are not known analytically, the linearised system remains non-local and perturbations must satisfy positivity and mass-preserving constraints, while also leaving the population-level age distribution unaffected. For these reasons, we reserve the stability analysis of these stationary distributions for future research.

4.4. Symmetry

In this section, we show that if $\mu (x)$ is symmetric around zero, then so is the steady state.

Let $\rho (a,x)$ be a solution to (13). Assume that for all $x\in [\! - \!1,1]$ that $\rho (0,x) = \mu (x) = \mu ({-}x)$ .

Define $\tilde {\rho }(a,\tilde {x}) = \rho (a,-x)$ . Then

\begin{align*} \partial _a \tilde {\rho }(a,x) &= \partial _a \rho (a,-x) \\[4pt] &= - \partial _x \left( \rho (a,-x) \left( \int _U \int _0^A M(a,b) \, \varphi (y + x)\, \rho (b,y) \, \text{d}b \, \text{d}y \, \right) \right) + \, \frac {\sigma ^2}{2} \partial _{xx} \rho (a,-x)\end{align*}

\begin{align*} &= - \partial _x \left( \rho (a,-x) \left( \int _U \int _0^A M(a,b) \, \varphi ({-}y + x) \, \rho (b,-y) \, \text{d}b \, \text{d}y \, \right) \right) + \, \frac {\sigma ^2}{2} \partial _{xx} \rho (a,-x) \\[4pt] &= - \partial _x \left( -\rho (a,-x) \left( \int _U \int _0^A M(a,b) \, \varphi (y-x)\, \rho (b,-y) \, \text{d}b \, \text{d}y \, \right) \right) + \, \frac {\sigma ^2}{2} \partial _{xx} \rho (a,-x) \\[4pt] &= - \partial _x \left( -\tilde {\rho }(a,\tilde {x}) \left( \int _U \int _0^A M(a,b) \, \varphi (y-x) \, \tilde {\rho }(b,y) \, \text{d}b \, \text{d}y \, \right) \right) + \, \frac {\sigma ^2}{2} \partial _{xx} \tilde {\rho }(a,\tilde {x}) \\[4pt] &= - \partial _{\tilde {x}} \left( \tilde {\rho }(a,\tilde {x}) \left( \int _U \int _0^A M(a,b) \, \varphi (y-x) \, \tilde {\rho }(b,y) \, \text{d}b \, \text{d}y \, \right) \right) + \, \frac {\sigma ^2}{2} \partial _{xx} \tilde {\rho }(a,\tilde {x}) \,. \end{align*}

The condition on $\mu$ also ensures that the boundary conditions are the same. Hence, the derivative (in $a$ ) of $\rho (a,x)-\rho (a,-x)$ is equal to zero everywhere in $U$ and so $ \rho (a,-x) = \tilde {\rho }(a,\tilde {x}) = \rho (a,x)$ , meaning that (13) preserves the symmetry around $x=0$ present in $\mu (x)$ . Hence, if $\mu (x)$ is symmetric, then the steady state is also symmetric.

5.1. Age transport and conservation of mass

Define the density

\begin{align*} \pi (t,a) = \int _U \rho (t,a,x) \, \text{d}x \end{align*}

which describes the total density of individuals with age $a$ (across all opinions), which formally satisfies

\begin{align*} \partial _t \pi (t,a) &= \int _U \partial _t \rho (t,a,x) \, \text{d}x \\[3pt] &= - \tau \partial _a \int _U \rho (t,a,x) \, \text{d}x - \int _U \partial _x F[\rho ](t,a,x) \, \text{d}x \\[3pt] &= - \tau \partial _a \pi (t,a) - F[\rho ](t,a,-1) + F[\rho ](t,a,1) \\[3pt] &= - \tau \partial _a \pi (t,a) \end{align*}

with the final equality holding due to the no-flux boundary conditions. Hence, $\pi$ satisfies the transport equation

\begin{equation*} \partial _t \pi (t,a) + \tau \partial _a \pi (t,a) = 0 , \end{equation*}

with periodic boundary condition $\pi (t,0) = \pi (t,1)$ arising from the age-zero boundary condition for $\rho$ . Therefore,

\begin{align*} \pi (t,a) = \pi (0, \{a - \tau t \} ) \end{align*}

where here $\{x\}$ denotes the fractional part of $x$ . Thus, if $\tau (0,a) = 1$ for all $a\in [0,1]$ , meaning there is initially a uniform age density, then this will also be preserved. In addition, we can also see that

\begin{align*} \int _U \int _0^A \rho (t,a,x) \, \text{d}a \, \text{d}x \end{align*}

is constant in time and so the total mass is conserved.

We note also that a similar argument to that in Section 4.4 can be applied to the full system to show that if $\rho (0,a,x)=\rho (0,a,-x)$ and $\mu (x) = \mu ({-}x)$ for all $a\in [0,1]$ and all $x\in [\! - \! 1,1]$ , then $\rho (t,a,x)=\rho (t,a,-x)$ for all $t \geq 0$ . That is, the PDE (10) preserves the symmetry around $x=0$ .

5.2. Evolution of the mean opinion

Based on the observation of a shifting consensus in Figure 1a, we next wish to consider how the mean opinion evolves in time. Define the mean opinion at age $a$ and the overall mean opinion, respectively, by

(21) \begin{align} m(t,a) &= \int _U x \, \rho (t,a,x) \, \text{d}x ,\\[-10pt]\nonumber \end{align}

(22) \begin{align} \bar {m}(t) &= \int _0^A m(t,a) \, \text{d}a = \int _0^A \int _U x \, \rho (t,a,x) \, \text{d}x \, \text{d}a \,. \end{align}

Then we have

\begin{align*} \frac{\text{d}\bar {m}}{\text{d}t} &= \int _0^A \int _U x \, \partial _t\rho (t,a,x) \, \text{d}x \, \text{d}a \\[4pt] &= - \int _0^A \int _U x [ \tau \partial _a\rho + \partial _x F[\rho ](t,a,x) ] \text{d}x \, \text{d}a \\[4pt] &= - \tau \int _U x \left( \int _0^A \partial _a\rho (t,a,x) \, \text{d}a \right) \text{d}x - \int _0^A \left( \int _U x \, \partial _x F[\rho ](t,a,x) \, \text{d}x \right) \, \text{d}a \\[4pt] &= - \tau \int _U x ( \rho (t,0,x) - \rho (t,1,x) ) \text{d}x - \int _0^A ( x \, F[\rho ](t,a,x) \Big |_{x=-1}^{x=1} - \int _U F[\rho ](t,a,x) \, \text{d}x ) \, \text{d}a \\[4pt] &= \tau ( m(t,0) - m(t,1) ) + \int _0^A \int _U F[\rho ](t,a,x) \, \text{d}x \, \text{d}a \,. \end{align*}

Note that the final equality holds as a result of the no-flux boundary conditions. Looking at the interaction term in the flux, we have

(23) \begin{align} \int _0^A \int _U & \rho (t,a,x) \bigg ( \int _U \int _0^A M(a,b) \, \varphi (y - x) \, \rho (t,b,y) \, \text{d}b \, \text{d}y \bigg ) \, \text{d}x \, \text{d}a \nonumber \\[4pt] &= \int _0^A \int _0^A \int _U \int _U \varphi (y - x) \, M(a,b) \, \rho (t,b,y) \, \rho (t,a,x) \, \text{d}x \, \text{d}y \, \text{d}a \, \text{d}b \end{align}

By comparing the effect of the two transformations $(x,a)\rightarrow ({-}x,a)$ , $(y,b)\rightarrow ({-}y,b)$ and $(x,a)\rightarrow ({-}y,b)$ , $(y,b)\rightarrow ({-}x,a)$ , we see that the integral (23) is equal to zero when $M(a,b) = M(b,a)$ for all $a,b\in [0,A]$ , meaning ages affect each other symmetrically. In this case, only the noise term remains in the flux and we obtain

(24) \begin{align} \frac{\text{d}\bar {m}}{\text{d}t} &= \tau \left( \int _U x\,\mu (x)\,\text{d}x - m(t,A) \right) + \frac {\sigma ^2}{2} \left( \int _0^A \rho (t,a,-1) \, \text{d}a - \int _0^A \rho (t,a,1) \, \text{d}a \right) \,. \end{align}

The first term describes evolution of the mean due to deaths/births, which brings the mean closer towards that of the age-zero distribution $\mu$ . The second term describes evolution of the mean due to interactions with the no-flux boundary conditions, which arise as individuals with opinions on the boundary can only move their opinions in one direction. These contributions help to explain the movement of clusters towards a more symmetric distribution when the mean of $\mu$ is zero, as is observed in Figure 2.

We next consider several scenarios in which a solution to the full problem (10) can be constructed from solutions of a different, typically simpler, system.

5.3. Convergence to consensus

We evaluate in a simple scenario how quickly the population converges to consensus. Assume $M\equiv 1$ and $\phi \equiv 1$ , so that all individuals interact equally regardless of age or opinion. Assume also that the initial distribution $\rho _0$ and the age-zero distribution $\mu$ are both symmetric. In this case, the PDE (10) reduces to

\begin{align*} \partial _t \rho (t,a,x) + \tau \partial _a\rho (t,a,x) + \partial _x \left( \rho (t,a,x) \left( \int _U \int _0^A (y - x) \, \rho (t,b,y) \, \text{d}b \, \text{d}y \, \right) \right) - \frac {\sigma ^2}{2}\partial _{xx}\rho (t,a,x) &= 0 \,. \end{align*}

We first calculate

\begin{align*} \int _U \int _0^A (y - x) \, \rho (t,b,y) \, \text{d}b \, \text{d}y &= \int _0^A \int _U y \, \rho (t,b,y) \, \text{d}y \, \text{d}b - x \, \int _U \int _0^A \rho (t,b,y) \, \text{d}b \, \text{d}y \\[5pt] &= \bar {m}(t) - x \,. \end{align*}

As $\rho _0$ and $\mu$ are both symmetric, the solution will remain symmetric and so $\bar {m}(t) = 0$ for all $t\geq 0$ . The PDE therefore reduces further to

(25) \begin{align} \partial _t \rho (t,a,x) + \tau \partial _a\rho (t,a,x) + \partial _x \big ( -x\,\rho (t,a,x) \big ) - \frac {\sigma ^2}{2}\partial _{xx}\rho (t,a,x) &= 0 \,. \end{align}

Note that we have now effectively decoupled each age, as interactions between different ages arose only via the mean $\bar {m}$ , which we know to be zero due to symmetry. Therefore, the only role played by age is to transport the density forward in $a$ . If we momentarily ignore this age transport term in (25), we obtain the Fokker-Planck equation of an Ornstein–Uhlenbeck process [Reference Gardiner20] (with reflecting boundary conditions). We use this connection to construct a solution to (25).

For a given density $\varrho (x)\in L^2(U)$ , let $\mathcal{S}_t [\varrho ](x)$ denote the weak solution at time $t\geq 0$ of the following initial-boundary value problem

(26a) \begin{align} \partial _t p(t,x) + \partial _x \big (\! - \! x\,p(t,a,x) \big ) - \frac {\sigma ^2}{2}\partial _{xx}p(t,a,x) &= 0 & \text{ in } U ,\\[-10pt]\nonumber \end{align}

(26b) \begin{align} x\,p(t,a,x) + \frac {\sigma ^2}{2}\partial _{x}p(t,a,x) &= 0 & \text{ on } \partial U = \{-1,1\} ,\\[-10pt]\nonumber \end{align}

(26c) \begin{align} p(0,x) &= \varrho (x) & \text{ in } U \,. \end{align}

We consider $\varrho \in L^2(U)$ as we will choose below $\varrho = \mu$ .

Then the solution to (25) is given by

(27) \begin{align} \rho (t,a,x) = \begin{cases} \mathcal{S}_{a\tau ^{-1}} [\mu ](x) & \text{ for } 0 \leq a \leq \tau t ,\\[5pt] \mathcal{S}_t [\rho _0(a-\tau t,\cdot )](x) & \text{ for } \tau t \leq a \leq A \,. \end{cases} \end{align}

For ‘large’ values of $a$ in the second case, $\rho (t,a,x)$ is given by the solution to (26) at the same time $t$ but applied to $\rho _0(a-\tau t,\cdot )$ to account for the age transport. For ‘smaller’ values of $a$ , $a-\tau t$ would be negative, meaning $\rho (t,a,x)$ is in fact the evolution of the age-zero distribution $\mu$ under (26) until time $a\tau ^{-1}$ (since it has been this length of time since this distribution was at $a=0$ ). Requiring $\rho _0(0,x) = \mu (x)$ ensures that there is no jump at $t = a\tau ^{-1}$ . Note that we can construct the solution in each section separately and concatenate them because the evolution at each age is independent.

For $t\geq \tau ^{-1} A$ all values of $a\in [0,A]$ fall into the first case. As $\mathcal{S}_a [\mu ](x)$ is independent of $t$ , the solution is therefore stationary for $t\geq \tau ^{-1} A$ , with the stationary distribution being entirely independent of the initial distribution $\rho _0$ .

Using this solution we can also examine the evolution of the opinion variance by studying

\begin{align*} v(t) = \int _U x^2 \, \mathcal{S}_t [\varrho ](x) \, \text{d}x \,. \end{align*}

Differentiating in time, then integrating by parts twice, we have

\begin{align*} \frac{\text{d}v}{\text{d}t} &= \int _U x^2 \, \partial _x \left( x \, \mathcal{S}_t [\varrho ](x) + \frac {\sigma ^2}{2} \partial _{x}\mathcal{S}_t [\varrho ](x) \right)\, \text{d}x ,\\[5pt] &= -2 \int _U x \, \left( x \, \mathcal{S}_t [\varrho ](x) + \frac {\sigma ^2}{2} \partial _{x}\mathcal{S}_t [\varrho ](x) \right)\, \text{d}x ,\\[5pt] &= -2 v(t) - \sigma ^2 \left( \mathcal{S}_t [\varrho ](1) - \mathcal{S}_t [\varrho ]({-}1) - \int _U \mathcal{S}_t [\varrho ](x) \, \text{d}x \right) \,. \end{align*}

As (26) also preserves symmetry, if $\varrho$ is symmetric then $\mathcal{S}_t [\varrho ](1) = \mathcal{S}_t [\varrho ]({-}1)$ . Moreover, the mass is preserved, so if $\varrho$ has mass 1, then so does $\mathcal{S}_t [\varrho ]$ . Note that both these properties are true of the choices for $\varrho$ in (27). In such a case, we conclude that $v(t)$ satisfies

\begin{align*} \frac{\text{d}v}{\text{d}t} &= \sigma ^2 -2v \quad \Rightarrow \quad v(t) = \frac {\sigma ^2}{2} + \bigg ( v(0) - \frac {\sigma ^2}{2} \bigg ) e^{-2t} \,. \end{align*}

Hence we conclude that the opinion density $\rho (t,a,x)$ has mean zero and variance

(28) \begin{align} \int _U x^2 \, \rho (t,a,x) \, \text{d}x = \begin{cases} \dfrac {\sigma ^2}{2} + \left( \displaystyle \int _U x^2 \, \mu (x) \, \text{d}x - \dfrac {\sigma ^2}{2} \right) e^{-2a\tau ^{-1}} & \text{ for } 0 \leq a \leq \tau t ,\\[10pt]\dfrac {\sigma ^2}{2} + \left( \displaystyle \int _U x^2 \, \rho _0(a - \tau t,x) \, \text{d}x - \dfrac {\sigma ^2}{2} \right) e^{-2t} & \text{ for } \tau t \leq a \leq A \,. \end{cases} \end{align}

Figure 5.

Opinion variance at each age over time, calculated from the numerical solution of (10) with $M \equiv 1$ , $\phi \equiv 1$ , $\tau = \sigma = 0.1$ and both $\rho _0$ and $\mu$ uniformly distributed. A white dashed line shows when $a = \tau t$ , which marks the transition between the two parts of the solution (27).

Figure 5 shows the opinion variance at each age over time, calculated from the numerical solution of 10 with $M \equiv 1$ , $\phi \equiv 1$ , $\tau = \sigma = 0.1$ and both $\rho _0$ and $\mu$ uniformly distributed. This matches exactly with the theoretical variance (28). Initially, there is a high variance at all ages due to the uniform initial condition, but as the population moves towards consensus, the only source of variance is from $\mu$ , which decays as individuals age.

5.4. No inter-age interactions

We now consider the other extreme and assume that the age-interaction kernel $M(a,b) = \delta _a(b)$ , that is, individuals only interact with others of exactly the same age. Note that this violates Assumption1 and is not a reasonable choice for the microscopic model as all individuals almost surely have distinct ages; therefore, there would simply be no opinion evolution. However, in the macroscopic setting, we will show that a solution for (10) can still be constructed.

For $M(a,b) = \delta _a(b)$ the flux becomes,

\begin{align*} F[\rho ](t,a,x) &= \rho (t,a,x) \bigg ( \int _U \, \varphi (y - x) \, \rho (t,a,y) \, \text{d}y\bigg ) - \, \frac {\sigma ^2}{2} \partial _{x}\rho (t,a,x) , \end{align*}

as each age $a$ only interacts with the density $\rho (t,a,x)$ at that same age. We therefore expect that the solution will match that of the standard mean-field model 11, transported forwards in age. We proceed in much the same way as in the previous Section 5.3.

For a given density $\varrho (x) \in L^2(U)$ , let $\mathcal{S}^{(2)}_t [\varrho ](x)$ denote the weak solution at time $t\geq 0$ of the classical mean-field limit, given by the following initial-boundary value problem

(29a) \begin{align} \partial _t p(t,x) + \partial _x G[p](t,x) &= 0\\[-10pt]\nonumber \end{align}

(29b) \begin{align} G[p](t,-1) = G[p](t,1) &= 0\\[-10pt]\nonumber \end{align}

(29c) \begin{align} p(0,x) &= \varrho (x) \end{align}

with flux term

(30) \begin{align} G[p](t,x) = p(t,x) \bigg ( \int _U \, \varphi (y - x) \, p(t,y) \, \text{d}y\bigg ) - \, \frac {\sigma ^2}{2} \partial _{x}p(t,x) \,. \end{align}

Then, similarly to the construction in Section 5.3, the solution to (10) for $M(a,b) = \delta _a(b)$ is given by

(31) \begin{align} \rho (t,a,x) = \begin{cases} \mathcal{S}^{(2)}_{a\tau ^{-1}} [\mu ](x) & \text{ for } 0 \leq a \leq \tau t ,\\[0.5em] \mathcal{S}^{(2)}_t [\rho _0(a-\tau t,\cdot )](x) & \text{ for } \tau t \leq a \leq A \,. \end{cases} \end{align}

Again, the distribution is stationary for $t\geq \tau ^{-1} A$ and independent of the initial distribution $\rho _0$ .

In both this example, where individuals interact only with those of exactly the same age, and the previous example of Section 5.3, where all individuals interact at all times, we exploit the decoupling of ages to construct the solution of (10) from the solution to simpler problems.

5.5. No ageing

In our next scenario, we consider how the full problem (10) can again be reduced in the case that $\tau = 0$ . While this may appear to remove the ageing effect entirely, it still allows for differing opinion distributions at different ages and certain non-trivial age interactions (specifically, we may still reduce the problem in the case that age-interactions have the form $M(a,b) = M(b)$ ).

Setting $\tau = 0$ in (10), we instead have

(32a) \begin{align} \partial _t \rho (t,a,x) + \partial _x F[\rho ](t,a,x) &= 0 \end{align}

with the same boundary conditions as in (10). Assume that $M(a,b) = M(b)$ , in which case the flux (9) reduces to

\begin{align*} F[\rho ](t,a,x) = \rho (t,a,x) \left( \int _U \varphi (y - x) \, \left( \int _0^A M(b) \, \rho (t,b,y) \, \text{d}b \right) \, \text{d}y \right) - \, \frac {\sigma ^2}{2} \partial _{x}\rho (t,a,x) \end{align*}

We first solve for

\begin{align*} u(t,x) = \int _0^A M(b) \, \rho (t,b,x) \, \text{d}b , \end{align*}

and use this to decouple the ages in (32). Formally, we have

\begin{align*} \partial _t u(t,x) &= \int _0^A M(b) \,\partial _t \rho (t,b,x) \, \text{d}b \\[4pt] &= -\int _0^A M(b) \, \partial _x \left( \rho (t,b,x) \left( \int _U \varphi (y - x) \, u(t,y) \, \text{d}y \right) \right) - \, M(b) \frac {\sigma ^2}{2} \partial _{xx}\rho (t,b,x) \, \text{d}b ,\\[4pt] &= - \partial _x \left( \int _0^A M(b) \, \rho (t,b,x) \left( \int _U \varphi (y - x) \, u(t,y) \, \text{d}y \right) \, \text{d}b \right) + \frac {\sigma ^2}{2} \partial _{xx} \bigg ( \int _0^A M(b) \rho (t,b,x) \, \text{d}b \bigg ) \,. \end{align*}

It is at this point crucial that $u(t,y)$ is independent of age, as we may now factorise the following integral,

\begin{align*} \int _0^A M(b) \, \rho (t,b,x) \bigg ( \int _U \varphi (y - x) \, u(t,y) \, \text{d}y \bigg ) \, \text{d}b = \left( \int _0^A M(b) \, \rho (t,b,x) \, \text{d}b \right) \left( \int _U \varphi (y - x) \, u(t,y) \, \text{d}y \right) \,. \end{align*}

If $M(a,b)$ depended on both $a$ and $b$ , rather than only on $b$ , we would require $u$ to be a function of $(t,a,x)$ and the above factorisation would not be possible. Using this factorisation, we conclude,

(33) \begin{align} \partial _t u(t,x) + \partial _x \left( u(t,x) \left( \int _U \varphi (y - x) \, u(t,y) \, \text{d}y \right) \right) - \frac {\sigma ^2}{2} \partial _{xx} u(t,x) = 0 , \end{align}

with appropriate no-flux boundary conditions. This is again the standard mean-field limit of the model without age structure 11. Using the notation of Section 5.4, we have

\begin{align*} u(t,x) = \mathcal{S}_t^{(2)}[u(0,\cdot )](x) = \mathcal{S}_t^{(2)}\left[\int _0^A M(b) \, \rho _0(a,\cdot )\right](x) \,. \end{align*}

Since $u(t,x)$ is now known, we may replace the flux term in (32) with

\begin{align*} F_u[\rho ](t,a,x) = \rho (t,a,x) \left( \int _U \varphi (y - x) \, u(t,y) \, \text{d}y \right) - \, \frac {\sigma ^2}{2} \partial _{x}\rho (t,a,x) \,. \end{align*}

Using this flux term, the evolution of $\rho (t,a,x)$ is now independent of the value of $\rho$ at any other age. Hence we can instead specify a family of problems: for each $a \in (0,1]$ find a solution $\rho ^{(a)}(t,x)$ of

(34a) \begin{align} \partial _t \rho ^{(a)}(t,x) + \partial _x F_u\big [\rho ^{(a)}\big ](t,x)&= 0\\[-10pt]\nonumber \end{align}

(34b) \begin{align} F_u\big [\rho ^{(a)}\big ](t,-1) = F_u\big [\rho ^{(a)}\big ](t,1) &= 0\\[-10pt]\nonumber \end{align}

(34c) \begin{align} \rho ^{(a)}(0,x) &= \rho _0(a,x) \end{align}

while for $a = 0$ we have the constant

\begin{align*} \rho ^{(0)}(t,x) &= \mu (x) \, \int _U \rho (0,1,y) \, \text{d}y. \end{align*}

Similarly to the fixed point argument used to show the existence of steady states in Section 4, we have again reduced the problem to a (family of) linear second-order parabolic PDE(s). Since the problems (34) differ only in their initial conditions $\rho _0(a,x)$ they will be continuous in $a$ if $\rho _0$ is, except at $a=0$ where the constant solution may create a discontinuity.

Note that a similar analysis is not possible when $\tau \gt 0$ since the evolution of $u$ would then be influenced by the boundary at $a=A$ . Specifically, it would require knowledge of the opinion density at age $A$ as this would be replaced continuously by $\mu$ as individuals die, in the same way that the mean opinion at age $A$ is continuously replaced by the mean of $\mu$ in (24).

5.6. McKendrick ageing

To conclude our analysis of the dynamics of (10), we now compare against the ageing structure utilised in the McKendrick (MK) equation, a commonly used model for age-structured population dynamics in the life sciences [Reference Inaba29, Reference Keyfitz and Keyfitz33, Reference M’kendrick38]. The MK equation allows for more complex population dynamics than the simple age-transport in (10); therefore, we are interested in cases where (10) can still incorporate some of these effects.

For a population of animals/individuals, the MK equation describes the density $\pi (t,a)$ of an age $a\in [0,\infty )$ at time $t\geq 0$ . The initial-boundary value problem reads

(35a) \begin{align} \partial _t \pi (t,a) + \tau \partial _a \pi (t,a) &= -\tau \, d(a) \, \pi (t,a) ,\\[-10pt]\nonumber \end{align}

(35b) \begin{align} \pi (t,0) &= \int _0^\infty d(a) \, \pi (t,a) \, \text{d}a ,\\[-10pt]\nonumber \end{align}

(35c) \begin{align} \pi (0,a) &= \pi _0(a) , \end{align}

where $d \, : \, [0,\infty )\rightarrow \mathbb{R}$ is the age-dependent death rate and $\pi _0\in L^\infty (\mathbb{R}^+)$ is the initial age profile. Note that in the standard MK model (11), the constant $\tau$ could be removed using a time rescaling; it is retained here to simplify the comparison with the opinion formation model (10). The boundary condition (35b) is chosen to ensure a constant total population size. In addition, we assume that

\begin{align*} \int _0^\infty \pi _0(a) \, \text{d}a = 1 , \end{align*}

so that

\begin{align*} \int _0^\infty \pi (t,a) \, \text{d}a = 1 , \end{align*}

for all $t\geq 0$ .

In the same way that (10) can be viewed as an extension of the age transport equation to include opinion dynamics, this setup can now be similarly extended to give

(36a) \begin{align} \partial _t \rho (t,a,x) + \tau \partial _a \rho (t,a,x) + \, \partial _x \tilde {F}[\rho ](t,a,x) &= -\tau \, d(a) \, \rho (t,a,x) ,\\[-10pt]\nonumber \end{align}

(36b) \begin{align} \tilde {F}[\rho ](t,a,-1) = \tilde {F}[\rho ](t,a,1) &= 0 ,\\[-10pt]\nonumber \end{align}

(36c) \begin{align} \rho (t,0,x) &= \mu (x) \int _U \int _0^\infty d(a) \, \rho (t,a,x) \, \text{d}a \, \text{d}x ,\\[-10pt]\nonumber \end{align}

(36d) \begin{align} \rho (0,a,x) &= \rho _0(a,x) , \end{align}

where the definition of $\tilde {F}[\rho ]$ is adapted slightly from (9) as

(37) \begin{align} \tilde {F}[\rho ](t,a,x) = \rho (t,a,x) \left( \int _U \int _0^\infty M(a,b) \, \varphi (y - x) \, \rho (t,b,y) \, \text{d}b \, \text{d}y - \, \frac {\sigma ^2}{2} \partial _{x}\log (\rho (t,a,x)) \right) \,. \end{align}

In addition, for a solution $\rho$ of (36), the age density

\begin{align*} \pi (t,a) = \int _U \rho (t,a,x) \, \text{d}x \end{align*}

satisfies the MK equation (35) with initial condition

\begin{align*} \pi _0(a) = \int _U \rho _0(a,x) \, \text{d}x , \end{align*}

and $\pi$ can thus be solved independently of $\rho$ (e.g. using the method of characteristics). Using this solution, we can replace or update the boundary condition (36c) to obtain

(38a) \begin{align} \partial _t \rho (t,a,x) + \tau \partial _a \rho (t,a,x) + \, \partial _x \tilde {F}[\rho ](t,a,x) &= -\tau \, d(a) \, \rho (t,a,x) ,\\[-10pt]\nonumber \end{align}

(38b) \begin{align} \tilde {F}[\rho ](t,a,-1) = \tilde {F}[\rho ](t,a,1) &= 0 ,\\[-10pt]\nonumber \end{align}

(38c) \begin{align} \rho (t,0,x) &= \mu (x) \left( \int _0^\infty d(a) \, \pi (t,a) \, \text{d}a \right) ,\\[-15pt]\nonumber \end{align}

(38d) \begin{align} \rho (0,a,x) &= \rho _0(a,x) \,. \end{align}

This approach allows for greater flexibility in specifying the population’s age profile, with the drawback that ages can become arbitrarily large. Indeed, steady states of the MK equation (35) have the form

\begin{align*} \pi (a) = \pi (0) \, \exp \bigg ( -\int _0^a d(b) \, \text{d}b \bigg ) \,. \end{align*}

So $\pi (a)$ will be compactly supported (on $[0,A]$ ) if and only if there exists some $A \gt 0$ such that

\begin{align*} \lim _{a \rightarrow A^-} d(a) = \infty , \end{align*}

that is, the death rate diverges. The following Proposition5.1, proven in Appendix A, shows that in such cases a solution to (38) can be constructed from a solution to (10). That is, the original model (10) with a uniform age profile can also capture the behaviour of the model (38), which uses the more realistic MK ageing.

Proposition 5.1. Assume that $\mu , d, \rho _0$ and $M$ are given and let $\pi (a)$ be the corresponding stationary age profile of ( 35 ), which is assumed to have compact support in $[0,1]$ . Let $q$ be a solution of ( 10 ) with age interactions given by $M'(a,b) = M(a,b) \, \pi (b)$ . Then $\rho$ defined by

(39) \begin{align} \rho (t,a,x) = q(t,a,x) \, \pi (a) \end{align}

is a solution of ( 38 ).

Therefore, by incorporating the stationary age profile $\pi$ into the age interaction term $M'$ , the original model (10) can capture some effects from the model using McKendrick ageing (38), in particular, a non-uniform age profile. It should be noted that this only applies in the case that $\pi$ is compactly supported, although in the context of opinion dynamics, this is a realistic assumption.

Note that if $M(a,b)$ is independent of $a$ then so is $M'(a,b)$ , hence Assumption2 still holds and the analysis of steady states in Section 4 is still applicable.

This raises the question of how a non-stationary age density $\pi (t,a)$ might affect the opinion formation process. For example, we observe in many numerical solutions to (10) that the population becomes increasingly clustered with higher ages. Therefore, a population with an increasing proportion of older individuals might become more influenced by these clusters, while a population with an increasing proportion of younger individuals might be increasingly influenced by the age-zero distribution $\mu$ .

This section has analysed several features of the macroscopic dynamics, establishing properties such as transport of the population-level age distribution, conservation of mass and mean dynamics. These provide some insight into the macroscopic behaviours observed in Section 3.2. We have also provided connections to several existing models, allowing the solution to (10) to be expressed in terms of solutions to other models in certain cases. These connections provide a possible avenue for further analysis. Finally, we have compared against the structure of the McKendrick equation for population dynamics, demonstrating when there is an equivalence between these approaches and when they differ, opening additional directions for future work.