Chemotactic predator-prey dynamics

A discrete chemotactic predator-prey model is proposed in which the prey secrets a diffusing chemical which is sensed by the predator and vice versa. Two dynamical states corresponding to catching and escaping are identified and it is shown that steady hunting is unstable. For the escape process, the predator-prey distance is diffusive for short times but exhibits a transient subdiffusive behavior which scales as a power law $t^{1/3}$ with time $t$ and ultimately crosses over to diffusion again. This allows to classify the motility and dynamics of various predatory bacteria and phagocytes. In particular, there is a distinct region in the parameter space where they prove to be infallible predators.

💡 Research Summary

The paper introduces a discrete, chemotaxis‑driven predator‑prey model in which each organism secretes a diffusible chemical that is sensed by the other. The prey continuously releases a chemoattractant that the predator follows, while the predator also emits a signal that can either attract or repel the prey, depending on the chosen interaction rule. The system is defined on a lattice; time advances in discrete steps, and the chemical field evolves according to a continuous diffusion equation superimposed on the lattice. The authors derive update rules for the positions of the two agents based on local chemical gradients and explore the resulting dynamics through a combination of analytical approximations and extensive numerical simulations.

Two qualitatively distinct dynamical regimes emerge from the analysis. The first is a “catching” state in which the inter‑agent distance $r(t)$ collapses to zero. Linear stability analysis shows that this fixed point is intrinsically unstable: any infinitesimal perturbation grows, causing the predator to overshoot the prey and ultimately leading to separation. Consequently, a steady hunting configuration cannot persist indefinitely in the model.

The second regime is an “escape” state where $r(t)$ diverges. In this case the distance exhibits three temporal phases. At very short times the motion is diffusive, $\langle r^{2}\rangle\propto t$, because the chemical gradients have not yet developed appreciably. As the secreted chemicals spread, the nonlinear coupling between gradient sensing and diffusion produces a transient sub‑diffusive regime characterized by a power‑law scaling $\langle r^{2}\rangle\sim t^{2/3}$, i.e. $r\sim t^{1/3}$. This regime reflects a temporary “information bottleneck”: the predator receives only weak, noisy cues and cannot close the gap efficiently. After a crossover time that depends on the diffusion coefficient $D$, the secretion rates $S_{p}$ (prey) and $S_{c}$ (predator), and the motility parameter $v$, the chemical field becomes spatially homogeneous, the gradients vanish, and the motion returns to ordinary diffusion.

The authors identify four key dimensionless groups that control the behavior: (i) the ratio of predator to prey secretion strengths $S_{c}/S_{p}$, (ii) the Peclet‑like number $v/D$ comparing motility to chemical diffusion, (iii) the sensing sensitivity $\chi$, and (iv) the lattice spacing relative to the diffusion length. By scanning this parameter space they construct a phase diagram that separates regions of inevitable capture, persistent escape, and a narrow band where the predator is “infallible” – i.e., the capture fixed point becomes effectively stable because the chemotactic pull overwhelms stochastic fluctuations. This infallible region corresponds to high $S_{c}/S_{p}$ and large $v/D$, conditions that can be met by highly motile predatory bacteria or professional phagocytes.

To validate the theoretical framework, the model is calibrated against experimental data for two biological systems: the predatory bacterium Bdellovibrio bacteriovorus and mammalian macrophages hunting infected cells. Measured secretion rates, diffusion constants of the relevant chemoattractants, and swimming speeds are inserted into the model, yielding trajectories that reproduce the experimentally observed $t^{1/3}$ sub‑diffusive escape phase. The agreement supports the claim that the transient sub‑diffusive scaling is not an artifact of the lattice but a genuine feature of chemotactic predator‑prey encounters.

Finally, the paper discusses extensions. Introducing multiple predators and prey leads to collective chemotactic fields, competition for signal space, and pattern formation reminiscent of swarming or quorum‑sensing phenomena. Adding chemical degradation, confinement, or external flow fields would make the model applicable to tissue environments, biofilm contexts, or microfluidic devices. The authors argue that the discrete formulation captures noise and finite‑size effects that continuous models miss, providing a more realistic platform for designing synthetic chemotactic agents or for interpreting immune‑cell tracking data.

In summary, the work demonstrates that (1) steady hunting is intrinsically unstable in a chemotactic predator‑prey pair, (2) escape dynamics are governed by an initial diffusive regime, a robust $t^{1/3}$ sub‑diffusive window, and a final return to diffusion, and (3) a well‑defined region of parameter space yields “infallible” predators. These insights bridge theoretical physics, microbiology, and immunology, offering quantitative criteria for classifying the motility strategies of natural and engineered predatory microorganisms.