Asymptotic Expansions and Sharp Decay Estimates for Multi-Order Tempered Fractional Cooperative Systems

Source abstract

We study the asymptotic behavior of multi-order fractional cooperative systems in which each component carries a Caputo tempered fractional derivative of its own order αi∈(0,1) with a common tempering parameter λ≥0. For cooperative vector fields that are homogeneous of degree p≥1 and admit a vector v≻0 with f(v)≺0, we prove that every solution starting in the nonnegative orthant is global, remains nonnegative, and satisfies 0≤Φi(t,ω)≤∥ω∥vvie−λt; for λ>0, this expresses decay at the rate of the free tempered evolution, that is, of the profile w0e−λt obtained when the tempered derivative vanishes identically, while for λ=0, it reduces to the known invariant-norm bound. When p=1, the estimate is sharpened to the tempered Mittag–Leffler rate e−λtEα̲(−ηtα̲), where α̲ is the smallest order. When p>1 and the tempering is strictly positive, λ>0, we derive the first-order asymptotic expansion eλtΦi(t,ω)=ωi+Ci(ω)tαi−1+o(tαi−1), in which Ci(ω) is given by an explicit convergent integral; hence, every component with positive initial value attains both the exact exponent and the exact amplitude of the free tempered profile, and a two-sided envelope for a scalar model problem, valid under an explicit smallness condition on the initial datum, makes this optimality quantitative for all times. As an application, we prove extinction at a tempered Mittag–Leffler rate for a class of tempered fractional Kolmogorov systems with net mortality. Numerical experiments with a predictor–corrector scheme support the theoretical findings.