Competition and Cooperation on Predation: Bifurcation Theory Of Mutualism Author: Srijana Ghimire Xiang-Sheng Wang University of Louisiana at Lafayette Introduction Existence and Stability of E1,E+ and E− Existence and property of Hopf bifurcation 3. R > 1 and R > 3R − 2R2. In this case, Q = Q , 1 2 1 1 c 1 points We investigate two predator-prey models which take into con- xc = 1/R1, E1 always exists, E1 is locally asymptotically H E + H E E sideration the cooperation between two different predators and stable if and only if Q < Q1, E− does not exist, and E+ exits + + R H within one predator species, respectively. Local and global dy- if and only if Q > Q1. 2.0 E E E - namics are studied for the model systems. By a detailed bi- - - Q Q Q Q Q Q furcation analysis, we investigate the dependence of predation + + h + h1 h2 1.5 no Hopf bifurcation Existence conditions of positive equilibria. (a) case 1(a) (b) case 1(b) (c) case 1(c) dynamics on mutualism (cooperative predation). H H E R + 2 E E two supercritical + + 1.0 R2 > 1 Q H E+ E - E Q1 - E y y - First Predator-Prey Model with Competition 1 E 1 E 1 1 y 1 E 1 0.5 one supercritical NA Q Q Q Q Q Q Q Q Q and Co-operation 1 + 1 + 1 h + h1 1 h2 R2 < one Q R1 NA if Q < Q1 (d) case 2(a) (e) case 2(b) (f) case 2(c) E± subcritical E+ if Q ≥ Q1 E+ d 0 H 0 2 4 6 8 E x = 1 − x − p xy − p xz − 2qxyz, (1) + 1 2 Q+ E Q1 + 0 Figure: Existence and property of Hopf bifurcation points in the (d, R) y = p xy + qxyz − d y, (2) 2 1 1 R2 = 3R1 − 2R NA 1 H parameter space. The red solid, blue dashed and black dotdashed curves are R y 0 1 1 y 2 2 p E 1 z = p xz + qxyz − d z, (3) 1 E R = d(d − 2)/(d − 1) , R = d(4 − d)/[5d − d − 2 + (d − 3) d(d − 3), 2 2 1 R2 = R1 Q Q Q Q and R = d/4, respectively. Where, x(t) is the ratio of prey density with respect to the 1 1 h1 h2 NA if Q < Q+ NA if Q < Q+ (g) case 3(a) (h) case 3(c) carrying capacity, y(t) and z(t) is the density of first kind and if if E± Q ≥ Q+ E± Q+ ≤ Q < Q1 Figure: E1(red),E+(red) and E−(green), with projection on y-Q plane if second kind of predators which compete for the prey x(t). E+ Q ≥ Q1 Existence and Stability of E+ and E− R Predation rate and death rate for each predator are pi and di 1 1.5 1 with i = 1, 2 respectively . In the absent of cooperative pre- Figure: Existence conditions of positive equilibria. Global Stability of E0 and E1 E E E + dation i.e. q = 0. The Gauss’s law of competitive exclusion + + H H principle holds.The basic reproduction numbers for the preda- E E - - E tors: R1 = p1/d1 and R2 = p2/d2. Stability of E+ Lyapunov function for the global stability of E0 : - Q Q Q Q Q V (x, y, z) = c (x − ln x + y + z) + (x + y + z − 1)2/2 + + h,+ + h,+ Lemma: Consider an ordinary differential system with a pa- 0 0 (a) case I.1 (b) case I.2 (c) case I.3 . E E Existence and Stability of E0,E1,E2,E+,E− rameter Q. Assume that an equilibrium E(Q) exists for Q in + + H Lyapunov function for the global stability of E1 : H E an interval I ⊂ R with characteristic polynomial p(ł,Q) = + 3 2 2 H Consider model (1)-(3) with R1 > R2. Denote Q = q/(d1d2) H ł + Q)a2(Q)ł + (Q)a1(Q)ł + Q)(Q)a0(Q). Assume that Q), V1(x, y, z) = c1(x−x1 ln x+y−y1 ln y+z)+(x+y+z−x1−y1) /2. 2 E and let s± is the root of f(s) = 2s − (Q + R1 + R2)s + Q = 0, (Q), a2(Q), a1(Q)¯and a0(Q) are all¯ positive and differentiable - Q Q Q 0 Q Q R1(R1−R2) Theorem: Assume R1 > R2 and the initial values + h,- h,+ 0 h,- h,+ Q1 = , ¯functions for Q ∈ I. Define g(Q) = a (Q)−a (Q)a (Q). Then √R1−1 0 1 2 (x(0), y(0), z(0)) are positive. If R ≤ 1 and q satisfies (d) case I.4 (e) case II.1 (f) case II.2 q 2 1 Q± = ( 2 ± 2 − (R1 + R2)) and the equilibrium E(Q) is locally asymptotically stable when g(Q) Figure: E+(blue) and E−(green), with projection on y-Q plane r 2 8 (d1+1) 2 is negative and unstable when g(Q) is positive. If g(Q) has a d1 + d2 + 2 d2[d1 − ] 2 <Q ,R ≥ 3R − 2R , 4(c0+1) d + 1 (d − 1) Q = 1 2 1 1 . q ≤ , c = max{ 2 , 1 }. c 2 simple root Qh ∈ I, then Hopf bifurcation occurs at Q = Qh 0 Q ,R ≤ 3R − 2R . 2c0 d2 − p2 4d1 : + 2 1 1 with crossing number Sign[g0(Q )]. Conclusion ¯ h Set xc = χ(Qc), Gc = G(xc) and G = maxx∈[0,x ] G(x), where , then E0 = (1, 0, 0) is globally asymptotically stable. If R1 > 1 c The stability condition of E+ is given in the following cases. the functions χ(Q) = x+ and and q satisfies 1. In both models cooperative predation may increase the survival a. G¯ < 0. In this case, E+ is always locally asymptotically stable. 2 2−R1x 2−R2x r 2 G(x) = 4x − (R1 + R2)x − − . (d1+1) probability of predators in the severe environment when non d2(1−R1x) d1(1−R2x) d1 + d2 + 2 d2[d1 − ] b. Gc > 0. In this case, G has a unique root, denoted by xh, on 4(c1/x1+1) The predator-free equilibrium E0 = (1, 0, 0) always exists, and q ≤ , cooperative predation is not efficient to battle with the natural (0, xc). E+ is locally asymptotically stable for Q ∈ (Qh, ∞) E is locally asymptotically stable if and only if R ≤ 1. 2c1x1 death. 0 1 and unstable for Q ∈ (Q ,Q ), where Q = φ(x ) is a Hopf 2 + h h h (d2 + 1)x1 + (d1 + d2)y1 x1(d1 − 1) The competitive-predation equilibrium E2 = (x2, 0, z2) with and, c = max{ , }. 2. Cooperative predation may destabilize a positive equilibrium bifurcation point. 1 d − p x 4d x2 = 1/R2 and z2 = 1/d2 − 1/p2 exists if and only if R2 > 1, 2 2 1 1 and induce a Hopf bifurcation.Depending on the model param- c. Gc < 0 and G¯ > 0. In this case, G has two roots, xh1 > and it is always unstable whenever it exists. , then E1 = (x1, y1, 0) is globally asymptotically stable. eters, the limit cycles bifurcated from the Hopf points may or x , on (0, x ). E is locally asymptotically stable for Q ∈ The existence conditions of another competitive-predation equi- h2 c + may not be stable. (Q ,Q )∪(Q , ∞) and unstable for Q ∈ (Q ,Q ), where librium E = (x , y , 0) with x = 1/R and y = 1/d − 1/p + h1 h2 h1 h2 1 1 1 1 1 1 1 1 Q = φ(x ) and Q = φ(x ) are two Hopf bifurcation Special Case and the cooperative-predation equilibria E = (x , y , z ) with h1 h1 h2 h2 ± ± ± ± points. References x± = 1/s±, y± = (s±−R2)/(d1Q) and z± = (s±−R1)/(d2Q). Biologically, we assume that the two predators can be regarded The stability conditions of E1 and E− are given in the following as the same. Thus, we can reduce the three-dimensional system • Beauchamp14 G. Beauchamp,Social Predation: How Group Living Benefits Predators cases. Direction of Hopf bifurcation and stability of and Prey, Academic Press, 2014. q (1)-(3) to a planar system 1. R ≤ 1. In this case, Q = Q , x = 2/Q , E does • Lo25 A. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, 1925. 1 c + c + 1 periodic solution x0(t) = 1 − x(t) − 2px(t)y(t) − 2qx(t)[y(t)]2, (4) • MTSS14 D. R. MacNulty, A. Tallian, D. R. Stahler, and D. W. Smith, Influence of group not exist, E exist if and only if Q ≥ Q , and E is always size on the success of wolves hunting bison, PLOS ONE 9 (2014), e112884. ± + − y0(t) = px(t)y(t) + qx(t)[y(t)]2 − dy(t). (5) unstable whenever it exists. Diagonalize the Jacobian matrix of (1)-(3). We calculate the • PR88 C. Packer and L. Ruttan, The evolution of cooperative hunting, Am. Nat. 132 (1988), 159–198. 2 Diagonalize the Jacobian matrix of (4)-(5). We calculate the 2. R1 > 1 and R2 ≤ 3R1 − 2R1. In this case, Qc = Q+, xc = first Lyapunov coefficient l1. The Hopf bifurcation is supercrit- • Sm95 H. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Com- q first Lyapunov coefficient l . The Hopf bifurcation is supercrit- petitive and Cooperative Systems, Mathematical Surveys and Monographs, vol. 41, Amer- 2/Q , E always exists, E is locally asymptotically stable ical when l1 < 0 and subcritical when l1 > 0 where, 1 + 1 1 ican Mathematical Society, Providence, RI, 1995.