Abstract
In the present paper, we study a class of nonlinear integro-differential equations of a kinetic type describing the dynamics of opinion for two types of societies: conformist ( σ = 1 ) and anti-conformist ( σ = − 1 ). The essential role is played by the symmetric nature of interactions. The class may be related to the mesoscopic scale of description. This means that we are going to statistically describe an individual state of an agent of the system. We show that the corresponding equations result at the macroscopic scale in two different pictures: anti-diffusive ( σ = 1 ) and diffusive ( σ = − 1 ). We provide a rigorous result on the convergence. The result captures the macroscopic behavior resulting from the mesoscopic one. In numerical examples, we observe both unipolar and bipolar behavior known in political sciences.
Highlights
In Ref. [1] a general model of swarming behavior of an individual population was proposed and studied
An arbitrary vector u ∈ Rd can be related to a biological state, activity, opinion, a social state of a test agent, etc.—cf. [1,5,6,7] and references therein, d = 1, 2, . . . . The model has a wide range of possible applications in various applied sciences, such as biology, medicine, social or political sciences
In the paper we studied a class of nonlinear integro-differential kinetic equations that can describe the opinion dynamics for two types of societies— the conformist σ = 1 and anti-conformist σ = −1
Summary
In Ref. [1] a general model of swarming behavior of an individual population was proposed and studied. [1] a general model of swarming behavior of an individual population was proposed and studied. The function f = f (t, u) is the distribution of an internal, microscopic state u ∈ Rd at time t ≥ 0 of a (statistical or test) agent Such a description has a mesoscopic nature. Because some results of the present paper may have a particular importance in the description in social sciences, we will refer the parameter u to as the opinion of (test) agent. Instead of dealing with the states of all separated agents involved in the population evolution, we deal with the statistical state of one (test ) agent This state is described by the probability density f that is a solution of Equation (1). The proof follows again by the standard application of Lipschitz-continuity in X (m)
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