Defect-mediated turbulence in the spatially extended Hastings–Powell period-doubling medium: pair-exchange mechanism, regimes beyond $b_1=3.5$, three-dimensional states, and basin structure

The spatially extended Hastings–Powell three-species food-chain reaction-diffusion system, whose local kinetics passes through a period-doubling (Feigenbaum) cascade to chaos as the parameter $b_1$ grows, supports a defect-induced turbulent state for $2.95 \le b_1 < 3.46$ with an unusual organization: the defect number saturates to a constant (no ongoing creation or annihilation), defects form charge-neutral bound pairs that collide and exchange constituents, and the pair-correlation function $g(r)$ exhibits an effective repulsive core. Purlis, Irurzun and Mola (Chem. Phys. Lett. 462 (2008) 64) mapped this phenomenology in one and two dimensions and described the defects as behaving like hard disks with translational degrees of freedom, resulting from a cooperative interaction between pairs of defects. Four linked questions remain open in the subsequent literature: (i) the dynamical mechanism and quantitative description of defect-pair formation, pair collisions, constituent exchange, and the microscopic origin of the repulsive core; (ii) the spatial structures and turbulent regimes for $b_1$ beyond the explored upper bound $b_1=3.5$; (iii) the existence and characterization of turbulent states in three dimensions; and (iv) the dependence of the final regime on initial conditions and the basin structure separating turbulent, spiral and no-wave attractors. These four questions remain unaddressed in the subsequent literature: the works that follow this study concern the same model family in a mixed-mode oscillatory regime, and the broader defect-turbulence literature concerns different media.

Background

A reaction-diffusion system couples local chemical or ecological kinetics to spatial transport, $\partial_t u = F(u) + D\nabla^2 u$, and in two dimensions such media generically host rotating spiral waves whose tips can destabilize into defect-mediated turbulence, a disordered state organized around continually interacting topological defects — point singularities where the phase of the local oscillation is undefined, operationally detected as intersections of isoconcentration contours. In excitable and simply periodic media this turbulence is classically a defect gas with continual creation and annihilation, with well-studied statistics (Hildebrand, Bär and Eiswirth, Phys. Rev. Lett. 75 (1995) 1503; Wang, Phys. Rev. Lett. 93 (2004) 154101) and with vortex-pair interactions understood within the complex Ginzburg–Landau amplitude-equation framework (Aranson and Kramer, Rev. Mod. Phys. 74 (2002) 99). In media whose local dynamics is more complex, new structures appear: synchronization and line defects in period-2 media (Goryachev, Chaté and Kapral, Phys. Rev. Lett. 80 (1998) 873 and Phys. Rev. Lett. 83 (1999) 1878), and defect-mediated turbulence driven by locally chaotic or quasiperiodic kinetics (Davidsen and Kapral, Phys. Rev. Lett. 91 (2003) 058303; Zhan and Kapral, Phys. Rev. E 73 (2006) 026224; Krefting and Beta, Phys. Rev. E 81 (2010) 036209). Whether a spiral destabilizes at all, and whether the resulting instability is absolute (growing at every point, so the whole wave breaks up) or merely convective, is organized by the absolute-versus-convective instability theory of Sandstede and Scheel (Phys. Rev. E 62 (2000) 7708). Purlis, Irurzun and Mola (Chem. Phys. Lett. 462 (2008) 64) studied the concrete case in which the local kinetics is the Hastings–Powell model (Hastings and Powell, Ecology 72 (1991) 896), a three-species food chain of resource $x$, consumer $y$ and predator $z$ with saturating type-II functional responses $f_1(x)=5.0\,x/(1+b_1 x)$ and $f_2(y)=0.1\,y/(1+2.0\,y)$, death rates $d_1=0.45$, $d_2=0.01$, and resource growth $x-x^2$. In its spatially extended version only the consumer and predator diffuse ($D_y=D_z=0.2$) under no-flux boundaries on lattices of side $L \ge 256$. As $b_1$ increases over $2.3 \le b_1 \le 3.5$ the local (space-free) dynamics follows a period-doubling cascade to chaos, and the medium realizes three regimes: decaying no-wave oscillations below $b_1=2.86$, rigidly rotating spirals for $2.86 \le b_1 < 2.95$, and defect-induced turbulence for $2.95 \le b_1 < 3.46$, with no-wave behavior again up to $b_1=3.5$. The spiral breaks up through an absolute instability at $b_1=2.95$ with no meandering anywhere in the range. The turbulent state is structurally unusual: after long transients the number of topological defects (intersections of the isoconcentration surfaces $x=x_0(b_1)$, $y=0.125$, $z=z_0(b_1)$, where $x_0(b_1)=\frac{b_1-1+\sqrt{(b_1-1)^2+1.5\,b_1}}{2b_1}$ is the positive root of $1-x-0.625/(1+b_1 x)=0$ and $z_0(b_1)=12.5\left(\frac{5x_0}{1+b_1 x_0}-0.45\right)$) saturates at a constant $N_{\mathrm{sat}}$ — so the statistically steady state has no defect creation or destruction — with density independent of system size and growing approximately linearly in $b_1$; the pair-correlation function $g(r)$ (the normalized density of defects at distance $r$ from a reference defect) shows a correlation hole, i.e. an effective repulsive core that shrinks as $b_1$ grows; and individual defects of topological charge $m=\pm1$ (defined by the rotation sense of the curled front around them) occur predominantly as charge-neutral bound pairs that collide with other pairs, exchange constituent defects, and thereby translate without changing the total defect number. In one dimension the analogous process, backfiring of traveling pulses (a pulse splitting into counterpropagating pulses), is only transient. This phenomenology differs from the creation–annihilation defect gas of excitable media and from line-defect turbulence; the source describes the defects as behaving like hard disks resulting from a cooperative interaction between pairs of defects, but a quantitative microscopic mechanism for pair formation, collision-mediated constituent exchange, and the effective repulsive core has not been established in the subsequent literature.

Problem Statement

Achieve a complete characterization of the defect-mediated turbulent state of the spatially extended Hastings–Powell system $\partial_t x = x - x^2 - \frac{5.0\,x}{1+b_1 x}\,y, \quad \partial_t y = \frac{5.0\,x}{1+b_1 x}\,y - \frac{0.1\,y}{1+2.0\,y}\,z - 0.45\,y + 0.2\,\nabla^2 y, \quad \partial_t z = \frac{0.1\,y}{1+2.0\,y}\,z - 0.01\,z + 0.2\,\nabla^2 z,$ with no-flux boundary conditions, whose local kinetics period-doubles to chaos as $b_1$ grows. Four linked questions, arising from that study and open in the subsequent literature, jointly define this characterization: (i) Mechanism. Provide a dynamical-mechanism description and its quantitative formulation of how defect pairs form, how pair–pair collisions proceed and exchange their constituent defects, and what microscopic interaction produces the effective repulsive core observed in the pair-correlation function $g(r)$; the mechanism must account for the constancy of the defect number in the statistically steady turbulent state ($2.95 \le b_1 < 3.46$) and for the shrinkage of the core with increasing $b_1$. (ii) Regimes beyond the explored range. Characterize the spatial structures and turbulent (or other) regimes for $b_1$ above the explored upper bound $b_1=3.5$. (iii) Three-dimensional behavior. Determine whether turbulent states exist in the three-dimensional extension of the same system, for which $b_1$ ranges and from which initial conditions, and characterize their statistics in terms of scroll filaments (the one-dimensional vortex cores of scroll waves, which are the three-dimensional analogues of spiral waves). (iv) Initial-condition dependence and basins. Map how the final regime depends on the initial state beyond the single fully developed spiral seed used in the source, and characterize the basin structure separating the turbulent, spiral and no-wave attractors. The four results jointly support one objective: the microscopic mechanism (i) explains the measured defect organization, while (ii)–(iv) establish the parameter range, the dimensional generality, and the multistability of that same turbulent state.

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