The Dynamics of a Holling-Tanner Predator-Prey Model with Allee Effect School of Mathematical Sciences Queensland University of Technology Claudio Arancibia-Ibarra, Michael Bode, Jose Flores, Graeme Pettet and Peter van Heijster SDG workshop - Blackheath Original Holling-Tanner Model [1] Nondimensionalisation & Desingularisation [2] Comparing the Dynamics Dynamical systems - theory and applications #5 - 2018 Ω = {(x, y) ∈ R 2 ,x> 0,y ≥ 0} (r, K, q, a, s, n) ∈ R 6 + The system is topologically equivalent to system A := a K < 1, S := s rK , Q := nq rK and M := m K ϕ(u, v, τ )= 0 @ Ku,nKv, u ⇣ u + a K ⌘ Kr 1 A =(x, y, t) du dτ = u 2 ((u + A) (1 - u)(u - M ) - Qv), dv dτ = Sv (u + A)(u - v). This set is an invariant region Future Work • Considering alternative food sources for the predator modifying the environmental carrying capacity. • Pattern formation (Turing and travelling waves) in spatial spatio-temporal extended Holling-Tanner models. Simulations of Strong and Weak Allee Effect Using System Parameters [3,4] rx ⇣ 1 - x K ⌘ ⇒ rx ⇣ 1 - x K ⌘ (x - m) dx dt = rx ⇣ 1 - x K ⌘ - qxy x + a dy dt = sy ⇣ 1 - y nx ⌘ References [1]E. Saez and E. Gonzalez-Olivares, Dynamics on a predator–prey model, SIAM Journal on Applied Mathematics, 59 (1999), pp. 1867–1878. [2]C. Arancibia-Ibarra, J. Flores, G. Pettet, and P. van Heijster, A Holling-Tanner predator-prey model with strong Allee efect, ArXiv e-prints, (2018), https://arxiv.org/abs/1809.05854. [3]I. Hanski, P. Turchin, E. Korpimaki, and H. Henttonen, Population oscillations of boreal rodents: regulation by mustelid predators leads to chaos, Nature, 364 (1993), p. 232. [4]P. Turchin and I. Hanski, An empirically based model for latitudinal gradient in vole population dynamics, The American Naturalist, 149 (1997), pp. 842–874. https://bat-smg.wikipedia.org/wiki/Abruozdielis:Mustela_1.jpg https://www.flickr.com/photos/peter-trimming/6038080307/in/photostream/ Φ = {(u, v), 0 ≤ u ≤ 1, 0 ≤ v ≤ 1} http://doi.org/10.6084/m9.figshare.7284557 http://doi.org/10.6084/m9.figshare.7284530