Preytaxis and Travelling Waves in an Eco-epidemiological Model

Abstract

Preytaxis is the attraction (or repulsion) of predators along prey density gradients and a potentially important mechanism for predator movement. However, the impact preytaxis has on the spatial spread of a predator invasion or of an epidemic within the prey has not been investigated. We investigate the effects preytaxis has on the wavespeed of several different invasion scenarios in an eco-epidemiological system. In general, preytaxis cannot slow down predator or disease invasions and there are scenarios where preytaxis speeds up predator or disease invasions. For example, in the absence of disease, attractive preytaxis results in an increased wavespeed of predators invading prey, whereas repulsive preytaxis has no effect on the wavespeed, but the wavefront is shallower. On top of this, repulsive preytaxis can induce spatiotemporal oscillations and/or chaos behind the invasion front, phenomena normally only seen when the (non-spatial) coexistence steady state is unstable. In the presence of disease, the predator wave can have a different response to attractive susceptible and attractive infected prey. In particular, we found a case where attractive infected prey increases the predators’ wavespeed by a disproportionately large amount compared to attractive susceptible prey since a predator invasion has a larger impact on the infected population. When we consider a disease invading a predator–prey steady state, we found some counter-intuitive results. For example, the epidemic has an increased wavespeed when infected prey attract predators. Likewise, repulsive susceptible prey can also increase the infection wave’s wavespeed. These results suggest that preytaxis can have a major effect on the interactions of predators, prey and diseases.

Appendices

Appendix A Analytic Minimum Wavespeeds

To find analytical minimum wavespeeds, we need to assume there is a travelling wave solution with constant wavespeed, with a native steady state ahead of the wave and a coexisting steady state behind the wave. By doing so, we use the transformation $Z=X-\omega T$ (where $\omega$ is the constant wavespeed) to arrive at a system of ODEs. After linearising ahead of the wave (i.e. at the native steady state) we look at the eigenvalues to see if there are any complex eigenvalues that would lead to unrealistic travelling waves (i.e. negative populations), and consequently find conditions on the wavespeed.

1.1 A.1 Calculating for Minimum Wavespeeds

For travelling waves, we will seek solutions of the form $(N(X,T),I(X,T),P(X,T))=(N(Z),I(Z),P(Z))$, where $Z=X-\omega T$ and $\omega$ is the wavespeed. We will also assume, for theoretical purposes, that the spatial domain is infinite. This is not a big assumption since the spatial domain is much larger than the wave itself. Likewise, we have assumed that $D_\mathrm{R}=1$. With this, the PDEs in Eqs. (10)–(12) become:

$$\begin{aligned} -\omega \frac{\mathrm{d} N}{\mathrm{d} Z}&=\frac{\mathrm{d}^2 N}{{\mathrm{d} Z}^2}+Ng(N)-\mu I-f(N,I)P, \end{aligned}$$

(13)

$$\begin{aligned} -\omega \frac{\mathrm{d} I}{\mathrm{d} Z}&=\frac{\mathrm{d}^2 I}{{\mathrm{d} Z}^2}+I(k(N,I)-f_\mathrm{I}(N,I)P), \end{aligned}$$

(14)

$$\begin{aligned} -\omega \frac{\mathrm{d} P}{\mathrm{d} Z}&=D_\mathrm{P}\frac{\mathrm{d}^2 P}{{\mathrm{d} Z}^2}-\frac{\mathrm{d}}{\mathrm{d} Z}\left( P F_\mathrm{S} \frac{\mathrm{d} N}{\mathrm{d} Z}+P (F_\mathrm{I}-F_\mathrm{S})\frac{\mathrm{d} I}{\mathrm{d} Z}\right) +P(f(N,I)-1). \end{aligned}$$

(15)

Equations (13)–(15) can be rewritten as a system of six first-order ODEs,

$$\begin{aligned} \frac{\mathrm{d}N}{\mathrm{d}Z}&={\dot{N}}, \end{aligned}$$

(16)

$$\begin{aligned} \frac{\mathrm{d}{\dot{N}}}{\mathrm{d}Z}&=-\omega {\dot{N}}-(Ng(N)-\mu I-Pf(N,I)), \end{aligned}$$

(17)

$$\begin{aligned} \frac{\mathrm{d}I}{\mathrm{d}Z}&={\dot{I}}, \end{aligned}$$

(18)

$$\begin{aligned} \frac{\mathrm{d}{\dot{I}}}{\mathrm{d}Z}&=-\omega {\dot{I}}-I(K(N,I)-Pf_{I}(N,I)), \end{aligned}$$

(19)

$$\begin{aligned} \frac{\mathrm{d}P}{\mathrm{d}Z}&={\dot{P}}, \end{aligned}$$

(20)

$$\begin{aligned} \frac{\mathrm{d}{\dot{P}}}{\mathrm{d}Z}&=\frac{-1}{D_\mathrm{P}}\left( \omega {\dot{P}}+P(f(N,I)-1) -\left( F_\mathrm{S}\left( {\dot{P}}{\dot{N}}+P\frac{\mathrm{d}{\dot{N}}}{\mathrm{d}Z}\right) +(F_\mathrm{I}-F_\mathrm{S})\left( {\dot{P}}{{\dot{I}}} +P\frac{\mathrm{d}{\dot{I}}}{\mathrm{d}Z}\right) \right) \right) \nonumber \\&=\frac{-1}{D_\mathrm{P}}[{\dot{P}}(\omega -F_\mathrm{S} {\dot{N}} - (F_\mathrm{I}-F_\mathrm{S}){\dot{I}})+ P(f(N,I)-1) \ldots \nonumber \\&\ldots -P[F_\mathrm{S}[-\omega {\dot{N}}-(Ng(N)-\mu I-Pf(N,I))] \ldots \nonumber \\&\ldots +(F_\mathrm{I}-F_\mathrm{S})[-\omega {\dot{I}}-I(K(N,I)-Pf_{I}(N,I))]]] \end{aligned}$$

(21)

Without any preytaxis ($F_\mathrm{S}=F_\mathrm{I}=0$), Eq. (21) becomes:

$$\begin{aligned} \frac{\mathrm{d}{\dot{P}}}{\mathrm{d}Z}=\frac{-1}{D_\mathrm{P}}(\omega {\dot{P}}+P(f(N,I)-1)). \end{aligned}$$

The Jacobian for Eqs. (16)–(21) (including preytaxis) is:

1.1.1 A.1.1 Predator Invasion in the Absence of Infected Prey

Consider that there is a prey-only steady state in front of a travelling wave of predators (thus we will ignore all infected prey equations/terms). We can linearise around this steady state, $(N,{\dot{N}},P,{\dot{P}})=(N^*,0,0,0)$, where $g(N^*)=0$, and ignore the disease, to get the Jacobian:

$$\begin{aligned} \left( \begin{array}{cccc} 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ cN^* &{}\quad -\omega &{}\quad \mu &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad -\frac{f(N^*,0)-1}{D_\mathrm{P}} &{}\quad -\frac{\omega }{D_\mathrm{P}} \\ \end{array} \right) \end{aligned}$$

Fortunately, this Jacobian is block upper triangular, so the eigenvalues are the eigenvalues of $\left( {\begin{matrix}0&{}1\\ cN^*&{}-\omega \end{matrix}}\right)$ and $\left( {\begin{matrix}0&{}1\\ -\frac{f(N^*,0)-1}{D_\mathrm{P}}&{}-\frac{\omega }{D_\mathrm{P}}\\ \end{matrix}}\right)$. The former has eigenvalues $\frac{-\omega \pm \sqrt{\omega ^2+4cN^*}}{2}$, which are always real, whereas the latter has eigenvalues $\frac{-\omega \pm \sqrt{\omega ^2-4D_\mathrm{P}(f(N^*,0)-1)}}{2D_\mathrm{P}}$, which are real as long as $\omega \ge 2\sqrt{D_\mathrm{P}(f(N^*,0)-1)}$. This means that the travelling wave has a wavespeed of at least $\omega _\mathrm{crit}=2\sqrt{D_\mathrm{P}(f(N^*,0)-1)}$, a minimum wavespeed that is the actual wavespeed if we assume ‘linear determinacy’. It is worth noting that this is independent of the preytaxis coefficients ($F_\mathrm{S}$ and $F_\mathrm{I}$). The reason for this is that at the leading edge of the predator invasion, prey density is nearly constant and thus there is no prey gradient for preytaxis to occur. However, this does not mean that preytaxis will have no effect on the wave away from the front edge.

1.1.2 A.1.2 Predator Invasion in the Presence of Susceptible and Infected Prey

Starting with the steady state $(N,{\dot{N}},I,{\dot{I}},P,{\dot{P}})=(N^*,0,I^*,0,0,0)$, where $N^*g(N^*)=\mu I^*$ and $K(N^*,I^*)=0$, the Jacobian becomes:

$$\begin{aligned} \left( \begin{array}{llllll} 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ -g(N^*)+cN^* &{}\quad -\omega &{}\quad \mu &{}\quad 0 &{}\quad f(N^*,I^*) &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ -I^*\frac{\partial K(N^*,I^*)}{\partial N} &{}\quad 0 &{}\quad -I^*\frac{\partial K(N^*,I^*)}{\partial I} &{}\quad -\omega &{}\quad I^* f_{I}(N^*,I^*) &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \frac{-1}{D_\mathrm{P}}(f(N^*,I^*)-1) &{}\quad \frac{-\omega }{D_\mathrm{P}} \end{array} \right) \end{aligned}$$

Again, this is block upper triangular, and thus we get the subsystem $\left( {\begin{matrix}0&{}1\\ -\frac{f(N^*,I^*)-1}{D_\mathrm{P}}&{}-\frac{\omega }{D_\mathrm{P}}\\ \end{matrix}}\right)$. This has eigenvalues $\frac{-\omega \pm \sqrt{\omega ^2-4D_\mathrm{P}(f(N^*,I^*)-1)}}{2D_\mathrm{P}}$. This means that the travelling wave has a wavespeed of at least $\omega _\mathrm{crit}=2\sqrt{D_\mathrm{P}(f(N^*,I^*)-1)}$, which is the wavespeed assuming ‘linear determinacy’.

The rest of the system is:

$$\begin{aligned} \left( \begin{array}{llllll} 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ -g(N^*)+cN^* &{}\quad -\omega &{}\quad \mu &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ -(\beta -c) I^* &{}\quad 0 &{}\quad \beta I^* &{}\quad -\omega \\ \end{array} \right) \end{aligned}$$

This subsystem has eigenvalues $\lambda =\frac{-\omega \pm \sqrt{\omega ^2-2(A \pm \sqrt{A^2-4B})}}{2}$, where $A=g(N^*)-cN^*-\beta I^*<0$ and $B=I^*(\beta (\mu +cN^*-g(N^*))-c\mu )>0$ (these are the trace and determinant of the Jacobian of the susceptible infected prey subsystem around the endemic steady state, and $A^2-4B<0$ is the condition for the steady state to be a stable focus). These eigenvalues, however, can have complex parts since $N^*,I^*>0$, and thus a focus around $(N^*,I^*)$ can be realistic (i.e. no issue about negative populations) and consequently this subsystem should not pose a restriction on the wave speed.

1.1.3 A.1.3 Disease Invasion in the Presence of Predators

Here, we start with the steady state $(N,{\dot{N}},I,{\dot{I}},P,{\dot{P}})=(N^*,0,0,0,P^*,0)$, where $f(N^*,0)=1$ and $P^*=N^*g(N^*)$ (and assuming $g(N^*)-cN^*-P^*\frac{\partial f(N^*,0)}{\partial N}<0$ for the steady state to be stable). Then the Jacobian becomes:

$$\begin{aligned} \left( \begin{array}{llllll} 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ -g(N^*)+cN^*+P^*\frac{\partial {f(N^*,0)}}{\partial N} &{}\quad -\omega &{}\quad \mu &{}\quad 0 &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad -(K(N^*,0)-P^*f_\mathrm{I}(N^*,0)) &{}\quad -\omega &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ \frac{-P^*}{D_\mathrm{P}}\left( \frac{\partial f}{\partial N}(N^*,0)+F_\mathrm{S}\Bigg (g(N^*)\right. &{}\quad \frac{-F_\mathrm{S}\omega P^*}{D_\mathrm{P}} &{}\quad \frac{-P^*}{D_\mathrm{P}}\left( \frac{\partial f}{\partial I}(N^*,0)+F_\mathrm{S}\left( \mu +\frac{\partial f}{\partial I}(N^*,0)\right) \right. &{}\quad \frac{(F_\mathrm{S}-F_\mathrm{I})\omega P^*}{D_\mathrm{P}} &{}\quad \frac{F_\mathrm{S}P^*}{D_\mathrm{P}} &{}\quad \frac{-\omega }{D_\mathrm{P}}\\ \left. \left. -cN^*-P^*\frac{\partial {f(N^*,0)}}{\partial N}\right) \right) &{}\quad &{}\quad +(F_\mathrm{I}-F_\mathrm{S})(K(N^*,0)-P^*f_\mathrm{I}(N^*,0))\Big )&{}\quad &{}\quad &{}\quad \\ \end{array} \right) \end{aligned}$$

The middle two rows (for I and ${\dot{I}}$) can be separated as all other terms in these rows are zero. Thus we have the matrix $\left( {\begin{matrix}0&{}1\\ -(K(N^*,0)-P^*f_\mathrm{I}(N^*,0))&{}-\omega \\ \end{matrix}}\right)$, which has the eigenvalues $\frac{-\omega \pm \sqrt{\omega ^2-4(K(N^*,0)-P^*f_\mathrm{I}(N^*,0))}}{2}$. These are real if $\omega ^2\ge 4(K(N^*,0)-P^*f_\mathrm{I}(N^*,0))$ and thus the suspected minimum wavespeed is $\omega _\mathrm{crit}=2\sqrt{K(N^*,0)-P^*f_\mathrm{I}(N^*,0)}$.

However, we need to check the other eigenvalues, namely of:

$$\begin{aligned} \left( \begin{array}{cccc} 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ -\pi &{}\quad -\omega &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ \frac{-P^*}{D_\mathrm{P}}\left( \frac{\partial f}{\partial N}(N^*,0) +F_\mathrm{S}\pi \right) &{}\quad \frac{-F_\mathrm{S}\omega P^*}{D_\mathrm{P}} &{}\quad \frac{F_\mathrm{S}P^*}{D_\mathrm{P}} &{}\quad \frac{-\omega }{D_\mathrm{P}} \\ \end{array} \right) \end{aligned}$$

(22)

where $\pi =g(N^*)-cN^*-P^*\frac{\partial f}{\partial N}(N^*,0)$. The eigenvalues of this system are difficult to find given this is a quartic equation. However, they do not need to be real as they represent the predator–prey subsystem and spiralling around the predator–prey steady state poses no threat of negative populations. Thus we do not have any more restrictions on the values for $\omega$.

However, if we assume that $F_\mathrm{S}=0$ and $D_\mathrm{P}=1$, then the characteristic equation can be reduced from a quartic to a quadratic equation: $\tau ^2+\pi \tau +P^*\frac{\partial f}{\partial N}(N^*,0)$, where $\tau =\lambda (\lambda +\omega )$ and $\pi =g(N^*)-cN^*-P^*\frac{\partial f}{\partial N}(N^*,0)<0$. From this, we have $\tau =\frac{-\pi \pm \sqrt{\pi ^2 -4P\frac{\partial f}{\partial N}(N^*,0)}}{2}$. Thus we have $\lambda =\frac{-\omega \pm \sqrt{\omega ^2+4\tau }}{2}$. Since $\pi <0$ and $\frac{\partial f}{\partial N}(N^*,0)>0$, then all eigenvalues are real if and only if $\tau$ is real, i.e. $\pi ^2>4P^*\frac{\partial f}{\partial N}(N^*,0)$. This condition is the same as the condition for the predator–prey steady state to be stable. The eigenvalues for other values of $F_\mathrm{S}$ and $D_\mathrm{P}$ have not been found.

1.2 A.2 Summary and Conclusions

To summarise the previous calculations, the analytical minimum wavespeeds are:

• Predator invasion in the absence of infected prey:$\omega _\mathrm{crit}=2\sqrt{D_\mathrm{P}(f(N^*,0)-1)}$, where $N^*$ is the density of prey at the disease-free prey-only steady state.

• Predator invasion in the presence of infected prey: $\omega _\mathrm{crit}=2\sqrt{D_\mathrm{P}(f(N^*,I^*)-1)}$, where $N^*$ and $I^*$ are the densities of the total prey and infected prey at the endemic prey-only steady state, respectively.

• Disease invasion in the presence of predators: $\omega _\mathrm{crit}=2\sqrt{K(N^*,0)-P^*f_\mathrm{I}(N^*,0)}$, where $N^*$ and $P^*$ are the densities of the prey and predator at the disease-free prey–predator steady state, respectively.

Appendix B Numerical Methods

The initial condition consists of two parts. First, there are the native specie(s), which we assume will be at the relevant (stable, at least in a non-spatial sense) coexistent steady state. The predator–prey and endemic prey steady-state initial conditions are derived by running MATLAB’s ‘ode45’ and taking their densities at the final time ($t=1000$). The invading initial condition will generally be a step function of 0.1 for $x\le 20$ and zero otherwise. However, for some scenarios, in particular when predator diffusion is very small (or zero), it is preferable for a smooth initial condition to be used. In these cases, a smooth approximation of the step function, $0.05(1-\tanh (x-20))$, is used.

The numerical scheme can be written as follows:

$$\begin{aligned} N(x,t+t_\mathrm{step})&=N(x,t)+N_\mathrm{growth}(x,(t,t+t_\mathrm{step}))+N_\mathrm{diffusion}(x,(t,t+t_\mathrm{step})), \end{aligned}$$

(23)

$$\begin{aligned} I(x,t+t_\mathrm{step})&=I(x,t)+I_\mathrm{growth}(x,(t,t+t_\mathrm{step}))+I_\mathrm{diffusion}(x,(t,t+t_\mathrm{step})), \end{aligned}$$

(24)

$$\begin{aligned} P(x,t+t_\mathrm{step})&=P(x,t)+P_\mathrm{growth}(x,(t,t+t_\mathrm{step})) +P_\mathrm{diffusion}(x,(t,t+t_\mathrm{step}))\ldots \nonumber \\&\ldots +P_\mathrm{taxis}(x,(t,t+t_\mathrm{step})). \end{aligned}$$

(25)

where, for example $N_\mathrm{growth}(x,(t,t+t_\mathrm{step}))$, is the growth of N at point x over the time interval $(t,t+t_\mathrm{step})$.

However, each of these terms have different properties. In particular, using one numerical scheme to deal with all these simultaneously would be highly problematic. In particular, the diffusion terms suggest using a scheme appropriate for parabolic PDEs, but such schemes would have real difficulty handling the taxis terms. Instead of trying to use one scheme to solve the whole system simultaneously, we will split the system into a sequence of smaller problems using a Strang splitting scheme (Chapter 18 of LeVeque 1992; Tyson et al. 2000). This scheme is implemented as follows.

First, solve the diffusion only problem numerically for half a time step and take this as the new solution at time t, i.e. for predators we have:

$$\begin{aligned} P^*(x,t):=P(x,t+0.5*t_\mathrm{step})=P(x,t)+P_\mathrm{diffusion}(x,(t,t+0.5t_\mathrm{step})) \end{aligned}$$

(26)

Do the same for susceptible and infected prey to derive $N^*(x,t)$ and $I^*(x,t)$, respectively. Following this, we then perform a taxis half step using an appropriate numerical scheme to again get a new solution at time t (note, this step only changes the predators since there is no taxis in the other classes).

$$\begin{aligned} P'(x,t):=P^*(x,t+0.5t_\mathrm{step})=P^*(x,t)+P_\mathrm{taxis}(x,(t,t+0.5t_\mathrm{step})) \end{aligned}$$

(27)

The next step is to take a full time step with only the growth dynamics, using an appropriate solver. This will form a new solution, which will be centred at time $t+0.5t_\mathrm{step}$.

$$\begin{aligned} {\hat{P}}(x,t+0.5t_\mathrm{step}):=P'(x,t+t_\mathrm{step})=P'(x,t)+P_\mathrm{growth}(x,(t,t+t_\mathrm{step})) \end{aligned}$$

(28)

Likewise, we get ${\hat{N}}(x,t+0.5t_\mathrm{step})$ and ${\hat{I}}(x,t+0.5t_\mathrm{step})$ by the same method, using $N'(x,t)$ and $I'(x,t)$ instead, respectively. Next, another taxis half step is taken (which only affects the predator equation). This gives a new solution at time $t+0.5t_\mathrm{step}$.

$$\begin{aligned} {\bar{P}}(x,t+0.5t_\mathrm{step}):={\hat{P}}(x,t+t_\mathrm{step}) ={\hat{P}}(x,t)+P_\mathrm{taxis}(x,(t+0.5t_\mathrm{step},t+t_\mathrm{step}))\nonumber \\ \end{aligned}$$

(29)

Finally, we take this solution and incorporate a half step of diffusion to get a final solution for time $t+t_\mathrm{step}$.

$$\begin{aligned} P(x,t+t_\mathrm{step}):={\bar{P}}(x,t+0.5t_\mathrm{step}) +P_\mathrm{diffusion}(x,(t+0.5t_\mathrm{step},t+t_\mathrm{step})) \end{aligned}$$

(30)

Do the same with ${\hat{N}}(x,t+0.5t_\mathrm{step})$ and ${\hat{I}}(x,t+0.5t_\mathrm{step})$ to get $N(x,t+t_\mathrm{step})$ and $I(x,t+t_\mathrm{step})$, respectively.

This scheme splits the problem into several smaller, more manageable steps, as well as allowing us to choose appropriate numerical methods for each subproblem instead of trying to use one scheme that would have difficulty handling the whole. One key advantage of this scheme is that it is of order 2 with respect to time and unconditionally stable as long as each subproblem is order 2 or higher.

For the growth step, the dynamics are local and thus a simple explicit ODEs solver can be used. We used the midpoint method (second-order Runge–Kutta). This is a reliable scheme for ODE, and because of this, it was chosen for the full step. For diffusion, both a forward-time–centred-space (FTCS) scheme and a Crank–Nicolson scheme were used and compared. The former is of order 1 with respect to time (order 2 with respect to space). This scheme is conditionally stable; it is stable if $\frac{t_\mathrm{step}}{(x_\mathrm{step})^2}<0.5$. The latter scheme is implicit and of order two with respect to time and space. It is unconditionally stable, although there are still numerical issues about artificial oscillations during the first few steps if $\frac{t_\mathrm{step}}{(x_\mathrm{step})^2}$ is too large and initial condition is too spiky. Consequently, the same step sizes will be used for both FTCS and Crank–Nicolson. Results between the two schemes have been compared and agree very well, the only visible difference being around $x=0$ in some cases of spatiotemporal chaos. There are no noticeable differences with respect to the wavespeed and the wavefront.

Fig. 13

Predator model. Attractive preytaxis, no predator diffusion. Step function as an initial condition (in contrast to Fig. 3 which used a smooth approximation). Dampened ‘spikiness’ occurs directly behind the wave due to the use of Lax-Wendroff to simulate preytaxis. However, comparing with Fig. 3, after some initial numerical issues around the discontinuity, the same travelling wave and wavespeed will eventually match, suggesting the numerical issue have little to no bearing on the long-term dynamics

For the taxis term, we have used a two-step Lax–Wendroff scheme. It is an explicit second-order (with respect to both time and space) scheme for hyperbolic PDEs (Chapter 11 of LeVeque 1992; Morton and Mayers 2002). It is very good at following suitably smooth solutions, but has issues around very large gradients and discontinuities, where solutions will overshoot and oscillate around sharp (i.e. non-smooth) points, particularly behind the discontinuity. These oscillations dampen away from the discontinuity. This can lead to issues in a few cases, especially if this results in negative populations. However, this scheme does follow the magnitude of peaks and their wavespeed very well, two key aspects to our exploration of travelling waves. Note that this issue only really matters if repulsive preytaxis is too strong compared to diffusion in predators. In particular, if $D_\mathrm{P}=0$, then the numerical scheme breaks down for any repulsive preytaxis as negative populations arise. Other numerical schemes were considered. For example, an upwind scheme was considered, but it is only of order 1 in time and space. It does not exhibit these oscillations around sharp points, but instead these points are smeared over as if there was some strong diffusive force, an effect that is undesirable as it would artificially flatten the wavefront, reducing prey gradients and thus reduce the strength of preytaxis. Another second-order scheme is Beam–Warming, which is like the Lax–Wendroff scheme except the oscillations are ahead of the wave (LeVeque 1992, Chapter 11). This potentially alters the dynamics ahead of the wave and may lead to negative populations in cases of attractive preytaxis. Likewise, the leapfrog scheme is of order two, but it has oscillations behind the wave that do not die out (Morton and Mayers 2002). These oscillations are generic for second-order schemes (LeVeque 1992, Chapter 11).

It is well worth noting that the inclusion of predator diffusion increases the smoothness of the numerical solutions, which improves the reliability of the Lax–Wendroff scheme. This is why the step function is used for every simulation where $D_\mathrm{P}>0$. However, in the absence of predator diffusion, Fig. 13a shows that the step function initial condition does not smooth out straight away but instead brings in damped oscillations/spikiness just behind the wavefront. This contrasts with Fig. 3a, where a smooth wave forms. However, even by $t=5$, the solution in Fig. 13a is largely smooth for the predator, that the oscillations and spikiness have largely gone. In fact, both Figs. 3b and 13b show that the wave moves at the same speed (actually, in Fig. 13a the travelling wave stays about 0.25 spatial units behind Fig. 3b over times $t=5,10,20,50\text { and }100$, this difference can be explained by the time taken to converge to the wavefront). However, without diffusion, even the slightest repulsive preytaxis results in negative predator populations around the discontinuity in the initial condition and the eventual breakdown of the numerical solution with Lax–Wendroff; in such cases, a Beam–Weaming scheme should be used instead as it would behave like Lax–Wendroff does in Fig. 13.

Boundary conditions are incorporated by setting the first and last spatial point to be equal to their immediate neighbour (and thus there is zero flux). This is done after each substep.

All simulations have step sizes of $t_\mathrm{step}=0.0005$ and $x_\mathrm{step}=0.05$ for time and space, respectively (unless stated otherwise). These step sizes satisfy stability conditions for the diffusion step as both FTCS and Crank-Nicolson because $\frac{t_\mathrm{step}}{(x_\mathrm{step})^2}=0.2<0.5$ (in fact, it probably is 0.1 due to the half steps). Other time and space step sizes have been considered and results do not look different as long as they are sufficiently small and the condition $\frac{t_\mathrm{step}}{(x_\mathrm{step})^2}<0.5$ is satisfied; in particular, the values for $t_\mathrm{step}$ and $x_\mathrm{step}$ in Fig. 4 give results that look identical to the default step sizes. This all suggests that the results in this paper are robust and a consequence of the system and not numerical artifacts.

Overall, the use of a Strang splitting scheme allows the full model to be split into a sequence of smaller models, which individually can be solved with well-established methods. Although there are some issues around a discontinuous initial condition for the taxis step, these issues are negligible with sufficient levels of diffusion for predators, and only seem to provide major inaccuracies when repulsive preytaxis is high and diffusion very small (in which case, the Beam–Weaming scheme is more appropriate).

Appendix C Non-constant Steady States of the Spatiotemporal Predator–Prey System

The purpose of this section is to demonstrate that attractive (repulsive) preytaxis has a (de)stabilising effect on constant steady states for the predator–prey system.

If we suppose that there is a (spatially varying) perturbation of the form $(N',P')\propto \exp (i\sigma x + \lambda t)$ around the constant coexistent predator– prey steady steady $(N^*,P^*)$. Then the Jacobian is:

$$\begin{aligned} J= \left( \begin{array}{c@{\quad }c} -\sigma ^2 + (g(N^*)-cN^*-P^*\frac{\partial f(N^*,0)}{\partial N}) &{} -1 \\ F_\mathrm{S}\sigma ^2 + P^*\frac{\partial f(N^*,0)}{\partial N} &{} -D_\mathrm{P} \sigma ^2 \\ \end{array} \right) \end{aligned}$$

(31)

For this Jacobian, it has eigenvalues, $\lambda$ that satisfy:

$$\begin{aligned} \lambda ^2 - \mathrm{trace}(J) \lambda + \det (J)=0. \end{aligned}$$

(32)

Now, $\mathrm{trace}(J)=-(1+D_\mathrm{P})\sigma ^2+(g(N^*)-cN^*-P^*\frac{\partial f(N^*,0)}{\partial N})$. A condition for stability would be that this is negative. If we assume that the steady state is stable in the absence of spatial effects, then $g(N^*)-cN^*-P^*\frac{\partial f(N^*,0)}{\partial N}<0$, then $\mathrm{trace}(J)<0$ for all $\sigma$.

Additionally, $\det (J)=D_\mathrm{P}\sigma ^2(\sigma ^2 - (g(N^*)-cN^*-P^*\frac{\partial f(N^*,0)}{\partial N}))+F_\mathrm{S}\sigma ^2+P^*\frac{\partial f(N^*,0)}{\partial N}$. Given $\mathrm{trace}(J)<0$, instability could occur if $\det (J)<0$ for some $\sigma ^2>0$, which is equivalent to there being a positive root (with respect to $\sigma ^2$) of $\det (J)=0$. Now, given $P^*\frac{\partial f(N^*,0)}{\partial N}>0$, both roots are of the same sign (and distinct), which means that these roots are positive (and thus instability can occur) if and only if $F_\mathrm{S}<D_\mathrm{P}(g(N^*)-cN^*-P^*\frac{\partial f(N^*,0)}{\partial N})$. However, $g(N^*)-cN^*-P^*\frac{\partial f(N^*,0)}{\partial N}$ is negative as a consequence of assuming a stable (non-spatial) steady state, meaning that the preytaxis has to be sufficiently repulsive ($F_\mathrm{S}\,{\ll }\, 0$) for instability to occur. Likewise, we can also conclude that attractive preytaxis ($F_\mathrm{S}>0$) has a stabilising effect.