Subcritical bifurcation of free elastic shell of biological cluster

In this paper we will investigate symmetry-breaking bifurcation of equilibrium forms of biological cluster. A biological cluster is a two-dimensional analogue of a gas balloon. The cluster boundary is connected with its kernel by elastic links. The inside part is filled with compressed gas or fluid. Equilibrium forms of biological cluster can be found as solutions of a certain second order ordinary functional-differential equation with four physical parameters: an elasticity coefficient α>0α>0 of boundary, an elasticity coefficient β>0β>0 of links and two parameters η,ν>0η,ν>0 describing compressed gas or fluid. For each multiparameter (α,β,η,ν)(α,β,η,ν) this equation possesses a radially symmetric solution. In Guze and Janczewska (2014) we proved the existence of symmetry-breaking bifurcation with respect to the ratio τ=β/ατ=β/α of elasticity coefficients. Now our aim is to describe bifurcation branches. Namely, applying a finite-dimensional reduction and a key function method we will prove the subcritical behaviour of biological cluster.