Short-wave signal versus indirect prey-taxis

The gist The study addresses a short-wave asymptotic for one class of quasi-linear second order PDE systems involving cross-diffusion described by the so-called Patlak–Keller–Segel law.

Indirect Prey-taxis Model Formulation

The research investigates the interaction between predators and prey where the predator's movement is driven by a signal produced by the prey, which is influenced by an external field. The system is described by a second-order PDE system governing the distributions of both species’ masses, involving kinetic terms for inter- and intra-specific reactions. Specifically, the equations read pt + div(χp∇s − δ1∇p) = pf(p, q), qt − div(µ∇q) = qg(p, q), st − div(δ2∇s) = κ2h + κ1q − νs (3). The driver’s intensity is an additional unknown regarding which there is an equation (3) describing its production by the prey influenced by the intensity of the external signal h.

Asymptotic Expansion and Homogenization

The study applies a short wave asymptotic to this multidimensional problem, given small diffusion rates δ > 0, setting h = h(x, t, x/δ, t/δ). The expansion is sought in the form (5), where the order of approximation K is as high as one wishes and the coefficients pk, sk are periodic in the fast coordinates. The analysis utilizes averaging operations such as (6) for averaging over fast variables, and partial averaging operations like (7) for spatial and temporal means.

Leading Approximation Results

When taking the limit as δ → +0, the system yields several key results. The solutions to equations (17) and (16) read s0 = ¯s(x, t), q0 = qξ0(x, t, τ). This leads to the relation (15), which makes equation (15) true identically. The leading approximation for the predator density is given by p0 = ¯pΦ(w0, µ1), where ¯p = ¯p(x, t) is nothing else than the mean predators density.

Slow Modulation and Stability Analysis

The analysis proceeds to a chain of equations for the coefficients pk−1, qk, sk that read (19)-(21). The leading approximation yields p0 = ¯pϕ∗, s0 = ¯s, q0 = ¯q. The system governing the slow modulation of the leading approximation turns out to be a byproduct of solving these equations for k = 2. This leads to a system that governs the slow modulation of the leading approximation, where p¯t + ∇· p¯∇s¯+ V(¯ ∇s¯) = ¯p f(¯p, q, ¯∇s̄), v̄j = (1/⟨eψj⟩−1e−1).

Effect of External Signal on Stability

The external signal influences the leading slow system by modifying both the kinetics and tactical transport of the predators. The averaged transport and kinetics get nonlocal. A notable exception occurs provided that the kinetic under consideration is p-linear, in which case ¯f = f(p, q) and ¯g = g(p, q). The analysis reveals that a generic stationary external signal generates the transport tensor that indicates suppressing the sensitivity. Moreover, there is a class of unsteady external signals, which enhance the sensitivity and taxical transport in certain directions.

Conclusion on Stability

The linear stability analysis reveals that an increase in the sensitivity leads to destabilization. The effect of the external signal is stabilizing when the drift suppresses the sensitivity to the driver’s intensity irrespective of the direction of its gradient and destabilizing when the sensitivity gets enhanced at least in some directions. The individual mode’s stability substantially depends on the direction of the wave vector, xˆ.

Discussion

The leading term in this asymptotic has the form of the short-wave package. Its amplitude undergoes a slow modulation, which, in turn, obeys the leading slow system written explicitly. The last one exhibits two main features caused only by the short waves: the drift of the predators and the modification of the system’s kinetics. The explicit expressions of the former reveal that an external signal is capable of suppressing the macroscopic mass transport.

Conclusions

The homogenization shows that a such a signal produces a drift (additional mass transport) and modifies the kinetic terms. The drift essentially depends on the signal’s shape. There are simple examples that unveil a substantial effect exerted from this drift on the taxical transport, but shaping the signal is capable of changing its effect to the opposite one. Regarding equilibria instabilities, the effect of the external signal is stabilizing when the drift suppresses the sensitivity to the driver’s intensity irrespective of the direction of its gradient and destabilizing when the sensitivity gets enhanced at least in some directions. There are examples of exponentially sharp stabilizing effects. The coexistence equilibrium is stable provided that supyˆ χdef = χ∗ χcr.

References

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[16] Tello I.J., Wrzosek D. (2016) Predator-prey model with diffusion and indirect prey-taxis.

[17] Tyutyunov Y., Titova L., Senina I.

1. Adaptive signal control for autonomous systems can be achieved by designing external signals that suppress or enhance taxical transport based on desired outcomes, such as suppressing macroscopic mass transport when a stationary signal is applied and enhancing it when a traveling wave is present.

2. AI systems can be designed to exhibit directional sensitivity changes in response to environmental inputs, as the analysis shows that the originally supposed isotropy of the predators sensitivity to the driving signal gets broken by wave propagation effects.

3. Stability analysis of complex multi-species or multi-agent systems can be performed using normal modes derived from a 3x3 matrix eigenvalue problem, which allows for determining if an equilibrium is stable or unstable based on the real part of eigenvalues.

4. AI models can predict pattern formation onset by identifying critical parameter sets where the neutral set boundary shifts, indicating transitions between stability and instability regions in the system's parameter space.

5. AI systems can learn to optimize control parameters by utilizing the calculated transport tensor, which measures the anisotropy of the predators sensitivity to the driver, enabling a determination of which direction is most dangerous for instability.

We address a short-wave asymptotic for one class of quasi-linear second-order PDE systems involving the cross-diffusion described by the so-called Patlak-Keller-Segel law. It is common to employ these equations for modeling the predator-prey community with the prey-taxis, which means the interactions of two species of particles or cells or anything else through which the species called "predators" is capable of moving directionally while searching for the other species called "prey." However, we suppose the predators to be sensitive not to the prey density but to a cue produced by the prey. Additionally, the production of the cue is assumed to be sensitive to the intensity of an external field, which is independent of the community state. This is what we call the external signal. It can be due to the spatiotemporal inhomogeneity of the environment arising from natural or artificial reasons. We assume that the external signal takes a general short-wave form and construct a complete formal asymptotic expansion for the short-wave solutions (modulo solving a chain of the so-called slow systems) with no restrictions on the spatial dimension or kinetics of inter- or intra-specific reactions. Further, we apply the short-wave asymptotic to studying the stability or instability induced by the external signal following Kapitza's theory for the upside-down pendulum. Applying the general results to some special classes of external signals, we get examples of suppressing the taxical transport, examples of the robustness of the species equilibrium to the signal up to a very strong stabilization or, oppositely, destabilization, and something like blurring the borderline in the parametric space between the areas of stability and instability of this equilibrium.