Chaotic dynamics in a simple predator-prey model with discrete delay

Figure 1.  Intersections of $ \theta(\tau)+2n\pi $ and $ \tau \omega_+(\tau), \ n = 0,1,\dots, $. Values of $ \tau $ at which the characteristic equation has pure imaginary eigenvalues, and hence candidates for critical values of $ \tau $ at which there could be Hopf bifurcations. In both graphs, at all such intersections, transversality holds, since the slope of these curves at these intersections are different. Parameters: $ m = 1,\ r = 1,\ K = 1, \ Y = 0.6 . $ (LEFT) $ s = 0.02 $. For $ n = 0 $ there are two intersections (i.e. $ j_0 = 2 $), at $ \tau_0^1 $ and $ \tau_0^2 $, but for $ n = 1 $, and hence $ n\geqslant 1 $, there are no intersections. (RIGHT) $ s = 0.007 $. There are two intersections each (i.e. $ j_n = 2, \ n = 0,1,2 $), at $ \tau_n^1 $ and $ \tau_n^2 $, for $ n = 0,1 $ and $ 2 $, but for $ n = 3 $, and hence $ n\geqslant 3 $, there are no intersections. In both (LEFT) and (RIGHT), $ E_+ $ is asymptotically stable for $ \tau\in[0,\tau_0^1)\cup(\tau_0^2,\tau_c) $ and unstable for $ \tau\in(\tau_0^1,\tau_0^2) $.

Figure 2.  Orbit diagrams. Initial data was taken to be $ x(t) = y(t) = 0.1 $ for $ t\in[-\tau,0] $. However, we found bistability in the portion of the diagram beween the vertical dots and varied the initial data as explained below. Except for the portion between the vertical dots, the rest of the diagram was the same for all of the initial conditions we tried (not shown). (TOP) All local maxima and minima for the $ y(t) $ coordinate of the attractor as $ \tau $ varies, including kinks. (BOTTOM) Diagram including local maxes and mins for the $ y(t) $ coordinate as $ \tau $ varies, but with kinks eliminated. There are two saddle-node of limit cycle bifurcations. They occur for $ \tau $ approximately equal to 76 and 82, where the curves in the orbit diagrams stop abruptly and there appear to be vertical dots. For $ \tau $ between these values, there is is bistability. Two orbitally asymptotically stable periodic orbits (with their maximum and minimum amplitudes shown) and an unstable periodic orbit with amplitudes between them (not shown). The two stable periodic orbits were found by producing this part of the orbit diagram varying $ \tau $ forward and then varying it backwards but startng at the last point of the attractor for the previous value of $ \tau $

Figure 3.  The time series for $ y(t) $ when $ \tau = 70 $, depicting kinks. There are two local maxima and two local minima over each period as shown in Figure 2 (TOP), but only one local maxima and one local minima in Figure 2 (BOTTOM) in which kinks have been removed

Figure 4.  Zoom-in of orbit diagram shown in Figure 2 for $ \tau\in[75,100] $ including kinks. The vertical dots indicate the boundary of the region of bistability, where the two saddle-node of limit cycle bifurcations occur

Figure 5.  Time delay embedding of two orbitally asymptotically stable periodic orbits demonstrating bistability for $ \tau = 81 $. The one with larger amplitude (dashed) has initial data $ x(t) = y(t) = 0.1 $, for $ t \in [-\tau,0] $, and period approximately $ 345 $. The one with smaller amplitude (solid) has initial data $ x(t) = y(t) = 0.1 $, for $ t \in [-\tau,0) $ and $ x(0) = 0.3, \ y(0) = 0.83 $ and has period approximately $ 273.7 $

Figure 6.  (LEFT) Time series starting from the initial data $ x(t) = y(t) = 0.1, t\in[-\tau,0] $ indicating how quickly the orbit gets close to the periodic attractor and (RIGHT) time delay embeddings of the periodic attractors, demonstrating the sequence of period doubling bifurcations initiating from the left at $ \tau\approx 83,\ 86 $, and $ 86.6 $. Values of $ \tau $ selected between these bifurcations: $ \tau = 82,\ 85,\ 86.3 $, and $ 86.8 $, with periods of the periodic attractor approximately equal to: $ 340.2,\ 564.6,\ 1144.5 $, and $ 2298.3 $, respectively, are shown

Figure 7.  (LEFT) Time series starting from the initial data $ x(t) = y(t) = 0.1, t\in[-\tau,0] $ indicating how quickly the orbit gets close to the periodic attractor and (RIGHT) time delay embeddings of the periodic attractors, demonstrating the sequence of period halfing bifurcations initiating from the left for values of $ \tau $ between $ 91.5 $ and $ 92 $, and at $ \tau\approx 92.2, \ 93.2, \ 98.3 $. Graphs shown are for values of $ \tau $ between these bifurcations: $ \tau = 91.95,\ 92,\ 93,\ 96 $, and $ 100 $, with periods approximately equal to: $ 4794.3, \ 2398.3, \ 1211.2,\ 557.6, $ and $ 295.3 $, respectively

Figure 8.  Time delay embedding starting at initial data $ x(t) = y(t) = 0.1, t\in[\tau,0] $ for $ \tau = 90.7 $ showing a periodic attractor with period approximately $ 1800 $, having $ 6(\neq2^n) $ loops for some integer $ n $

Figure 9.  (LEFT) Time series for $ \tau = 90 $ starting from the initial data $ x(t) = y(t) = 0.1, t\in[-\tau,0] $. (RIGHT) Time delay embedding of the strange attractor for $ \tau = 90 $. Only the portion of the orbit from $ t = 240,000-260,000 $ is shown

Figure 10.  The strange attractor, for $ \tau = 90 $, shown in Figure 9 in $ (x,y) $-space. Only the portion of the orbit from $ t = 240,000 $ to $ 260,000 $ is shown

Figure 11.  The return map for $ \tau = 90 $, computing the minimum value of $ y(t) $ as a function of the preceding minimum value of $ y(t) $ for $ y(t)<0.7 $ in both cases, using the data in Figure 9

Figure 12.  Time series (LEFT) for the solution that converges to a periodic attractor when $ \tau = 92 $, and (RIGHT) for the solution that converges to a strange attractor when $ \tau = 90 $, demonstrating that there is no sensitivity to initial data in the former case, but that there is sensitivity in the latter case. Initial data used for the solid curves: $ x(t) = y(t) = 0.1 $ for $ t\in[-\tau,0] $, and for the dotted curves: $ x(t) = 0.11 $ and $ y(t) = 0.1 $ for $ t\in[\tau,0] $