Lyapunov–Schmidt and Centre Manifold Reduction Methods for Nonlocal PDEs Modelling Animal Aggregations

Abstract

[Date] Cells adhere to each other and to the extracellular matrix (ECM) through protein molecules on the surface of the cells. The breaking and forming of adhesive bonds, a process critical in cancer invasion and metastasis, can be influenced by the mutation of cancer cells. In this paper, we develop a nonlocal mathematical model describing cancer cell invasion and movement as a result of integrin-controlled cell-cell adhesion and cell-matrix adhesion, for two cancer cell populations with different levels of mutation. The partial differential equations for cell dynamics are coupled with ordinary differential equations describing the extracellular matrix (ECM) degradation and the production and decay of integrins. We use this model to investigate the role of cancer mutation on the possibility of cancer clonal competition with alternating dominance, or even competitive exclusion (phenomena observed experimentally). We discuss different possible cell aggregation patterns, as well as travelling wave patterns. In regard to the travelling waves, we investigate the effect of cancer mutation rate on the speed of cancer invasion.

Kinetic Models for Pattern Formation in Animal Aggregations: A Symmetry and Bifurcation Approach

“…Since model ( 5) is nonlocal, next we confirm that the integrals (8) are welldefined for u ± satisfying conditions (9). We reproduce the calculation found in [13].…”

“…For a more in-depth review of pattern formation in hyperbolic models in biology, and the analytical and numerical techniques available to investigate them, see [21,59]. Existence of reduction methods (e.g., Centre Manifold reduction) for local bifurcations of various types of equations described in this section has been established for parabolic equations [31], for equations such as (1) in [13] and for hyperbolic age-structured models [44].…”

“…The convergence of Krylov subspace methods depends on the spectrum of the invariant solution under consideration. For equilibrium solutions, we know that this spectrum consists of isolated eigenvalues of finite multiplicity and that there are no accumulation points [13].…”

The impact of environmental noise on animal communication: pattern formation in a class of deterministic and stochastic hyperbolic models for self-organised biological aggregations

“…Throughout this study, we focused on the numerical investigation of the noise and its effects on various patterns (and transitions between patterns). We note here that transitions between the deterministic patterns can be investigated analytically with the help of bifurcation theory (see, for example, [22], [39], [40], [41] for the application of the equivariant bifurcation theory to the classification and investigation of patterns exhibited by nonlocal hyperbolic models (1)). Given the complexity of the nonlocal models (1) the application of this theory (which involves perturbations of the spatial states u ± (x, t)) is not very straightforward.…”

Fredholm property of nonlocal problems for integro-differential hyperbolic systems

“…Establishing a Fredholm property is a first step in developing a theory of local smooth continuation [12] and bifurcation [1,2,11] for Fredholm hyperbolic operators, in particular, such tools as Lyapunov-Schmidt reduction. Buono and Eftimie [1] consider autonomous 2×2 nonlocal hyperbolic systems in a single space variable, describing formation and movement of various animal, cell and bacterial aggregations, with some biologically motivated integral terms in the differential equations. One of the main results in [1] is a Fredholm alternative for the linearizations at a steady-state, which enables performing a bifurcation analysis by means of the Lyapunov-Schmidt reduction.…”