Apex Consumers and Trophic Downgrading Andrea de Lima Ribeiro (Oc.) Carlos Andrés Marcelo-Serván (Biol.) Carolina Arruda Moreira (Phys.) Cecilia Andreazzi (Biol.) César Parra (Phys.) Henrique Rubira (Undergrad.) II Southern-Summer School on Mathematical Biology Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 1 / 16
Some concepts • Apex (alpha, top-level) predators: no predators at their own; • Apex consumers may be able to model an ecosystem [1]; • Concept of trophic cascades; • Propagation of impacts by consumers on their prey downward through food webs. [3] Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 2 / 16
Some concepts • Loss of Apex changes the species composition and density of lower levels; • Top-down control is not the only process that modulate the species diversity. Figura: From [2] Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 3 / 16
The system Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 4 / 16
Populations Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 5 / 16
The article Ecological Applications, 14(5), 2004, pp. 1566–1573 q 2004 by the Ecological Society of America FISHING FOR LOBSTERS INDIRECTLY INCREASES EPIDEMICS IN SEA URCHINS K EVIN D. L AFFERTY 1 USGS Western Ecological Research Center, c/o Marine Science Institute, University of California, Santa Barbara, California 93106 USA Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 6 / 16
Objectives and questions Objectives To develop a mathematical model that describes the dynamics of the system that Lafferty (2004) studied. Questions • Is there evidence for top-down control? What are the conditions for its occurrance? • How do trophic cascade and epidemics interact? • How impacts on the apex predator population affects the food chain and epidemiological dynamics of this system? Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 7 / 16
Mathematical Model - Equations dK dt = K ( 1 − K ) − a ( U 1 + U 2 ) K dU 1 U 1 = − b L − m 0 U 1 − gm 1 U 1 + rU 1 U 2 dt U 1 + U 2 + U c dU 2 U 2 = ac ( U 1 + U 2 ) K − b L − m 0 U 2 − rU 1 U 2 dt U 1 + U 2 + U c dL U 1 + U 2 dt = bd L − m 2 L U 1 + U 2 + U c K , U 1 , U 2 , L : populations of kelps, infected urchins, not infected urchins and lobsters a , b : attack rates of urchins and lobsters c , d : efficiency of predation by urchins and lobsters r : transmission rate of infection m 0 , m 1 , m 2 : mortality rates of urchins (natural causes), urchins (infection) and lobsters g : multiplying factor U c : urchin critical density Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 8 / 16
Mathematical Model - Equations g ( U ) = 1 + α 1 − e − β ( U − U c ) 1 + e − β ( U − U c ) g(U) 1.5 1.0 0.5 Uc U α = 1 . 0 β = 5 . 0 Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 9 / 16
Results Sub-system: kelps and not infected urchins dK dt = K ( 1 − K ) − aU 2 K dU 2 = acU 2 K − m 0 U 2 dt Important fixed points: 1 K ∗ = 1 , U 2 ∗ = 0 : stable for m 0 > ac 2 K ∗ = m 0 ac , U 2 ∗ = ac − m 0 : exists and it’s stable for m 0 < ac a 2 c Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 10 / 16
Results a = 1 . 0 , c = 0 . 3 , m 0 = 0 . 5 1.0 0.8 K(t) 0.6 0.4 NI (t) 0.2 t 14. 28. 42. 56. 70. Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 10 / 16
Results a = 1 . 0 , c = 0 . 3 , m 0 = 0 . 5 NI 0.4 0.3 0.2 0.1 K 0.2 0.4 0.6 0.8 1.0 Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 10 / 16
Results a = 1 . 0 , c = 0 . 3 , m 0 = 0 . 03 1.0 0.8 K(t) 0.6 0.4 NI(t) 0.2 t 20. 40. 60. 80. 100. Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 10 / 16
Results a = 1 . 0 , c = 0 . 3 , m 0 = 0 . 03 NI 1.0 0.8 0.6 0.4 K 0.1 0.2 0.3 0.4 Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 10 / 16
Results Sub-system: kelps, infected and not infected urchins (lobsters have been removed) dK dt = K ( 1 − K ) − a ( U 1 + U 2 ) K dU 1 = − m 0 U 1 − gm 1 U 1 + rU 1 U 2 dt dU 2 = ac ( U 1 + U 2 ) K − m 0 U 2 − rU 1 U 2 dt Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16
Results Important fixed points: : stable for r < a 2 c ( m 0 + gm 1 ) 1 K ∗ = m 0 ac , U 1 ∗ = 0 , U 2 ∗ = ac − m 0 a 2 c ac − m 0 2 1 K ∗ = 2 acr [ r ( ac + m 0 + gm 1 ) � � r ( r ( ac + m 0 + gm 1 ) 2 + 4 ac ( m 0 + gm 1 )( − r + agm 1 )) − 1 ∗ = r ( ac − m 0 − gm 1 ) − 2 a 2 c ( m 0 + gm 1 ) U 1 � 2 a 2 cr � � r ( r ( ac + m 0 + gm 1 ) 2 + 4 ac ( m 0 + gm 1 )( − r + agm 1 )) + ∗ = m 0 + gm 1 U 2 r exists and it’s stable for r > a 2 c ( m 0 + gm 1 ) ac − m 0 Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16
Results r = 0 . 1 , α = 1 . 0 , m 1 = 0 . 05 , U c = 0 . 5 1.0 0.8 K(t) 0.6 I(t) 0.4 NI(t) 0.2 100. t 20. 40. 60. 80. Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16
Results r = 0 . 1 , α = 1 . 0 , m 1 = 0 . 05 , U c = 0 . 5 I 0.10 0.08 0.06 0.04 0.02 K 0.05 0.10 0.15 Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16
Results r = 0 . 1 , α = 1 . 0 , m 1 = 0 . 05 , U c = 0 . 5 NI 1.00 0.95 0.90 0.85 0.80 0.75 K 0.05 0.10 0.15 Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16
Results r = 0 . 1 , α = 1 . 0 , m 1 = 0 . 05 , U c = 0 . 5 NI 1.00 0.95 0.90 0.85 0.80 0.75 I 0.02 0.04 0.06 0.08 0.10 Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16
Results r = 0 . 5 , α = 1 . 0 , m 1 = 0 . 05 , U c = 0 . 5 1.0 K(t) 0.8 I(t) 0.6 NI(t) 0.4 0.2 100. t 20. 40. 60. 80. Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16
Results r = 0 . 5 , α = 1 . 0 , m 1 = 0 . 05 , U c = 0 . 5 I 0.5 0.4 0.3 0.2 0.1 K 0.05 0.10 0.15 0.20 0.25 0.30 Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16
Results r = 0 . 5 , α = 1 . 0 , m 1 = 0 . 05 , U c = 0 . 5 NI 1.0 0.8 0.6 0.4 0.2 K 0.05 0.10 0.15 0.20 0.25 0.30 Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16
Results r = 0 . 5 , α = 1 . 0 , m 1 = 0 . 05 , U c = 0 . 5 NI 1.0 0. 8 0.6 0.4 0.2 I 0 1 0 2 0 3 0 4 0 5 Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16
Results Effect of increased mortality: r = 0 . 5 , α = 5 . 0 , m 1 = 0 . 05 , U c = 0 . 5 1.0 K(t) 0.8 I(t) 0.6 NI(t) 0.4 0.2 t 20 40 60 80 100 Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16
Results All four populations (reintroduction of lobsters) dK dt = K ( 1 − K ) − a ( U 1 + U 2 ) K dU 1 U 1 = − b L − m 0 U 1 − gm 1 U 1 + rU 1 U 2 dt U 1 + U 2 + U c dU 2 U 2 = ac ( U 1 + U 2 ) K − b L − m 0 U 2 − rU 1 U 2 dt U 1 + U 2 + U c dL U 1 + U 2 dt = bd L − m 2 L U 1 + U 2 + U c Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16
Results Important fixed points: 1 1 K ∗ = 2 acr [ r ( ac + m 0 + gm 1 ) � � − 2 acr 2 ( m 0 + gm 1 ) + r 2 ( m 0 + gm 1 ) 2 + a 2 cr ( cr + 4 gm 1 ( m 0 + gm 1 )) − 1 acr − 2 a 2 c ( m 0 + gm 1 ) − r ( m 0 + gm 1 ) U 1 ∗ = � 2 a 2 cr � � − 2 acr 2 ( m 0 + gm 1 ) + r 2 ( m 0 + gm 1 ) 2 + a 2 ( 4 crgm 0 m 1 + cr ( cr + 4 g 2 m 12 )) + U 2 ∗ = m 0 + gm 1 r L ∗ = 0 stable for b < b 0 a 2 cU c b 0 = m 2 � � 1 + d ac − m 0 Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16
Results Important fixed points: 2 K ∗ = 1 − am 2 U c bd − m 2 U 1 ∗ = 0 m 2 U c U 2 ∗ = bd − m 2 1 L ∗ = � ( ac − m 0 )( bd − m 2 ) − a 2 cm 2 U c � ( bd − m 2 ) 2 dU c exists for b > b 0 and it’s stable for b 0 < b < b 1 with stable limit cycle for b > b 1 a 2 cU c b 0 = m 2 � � 1 + d ac − m 0 b 1 : no analytical expression Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16
Results b = 0 . 05 , d = 0 . 05 , m 2 = 0 . 01 0.6 K(t) 0.5 I(t) 0.4 0.3 NI(t) 0.2 L(t) 0.1 t 60 120 180 240 300 Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16
Results b = 0 . 05 , d = 0 . 05 , m 2 = 0 . 01 NI 0.6 0.5 0.4 0.3 0.2 0.1 K 0.05 0.10 0.15 0.20 0.25 0.30 Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16
Results b = 0 . 05 , d = 0 . 05 , m 2 = 0 . 01 NI 0.6 0.5 0.4 0.3 0.2 0.1 I 0.1 0.2 0.3 0.4 0.5 Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16
Results b = 0 . 05 , d = 0 . 05 , m 2 = 0 . 01 L 0.10 0.08 0.06 0.04 0.02 I 0.1 0.2 0.3 0.4 0.5 Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16
Results b = 0 . 05 , d = 0 . 05 , m 2 = 0 . 01 L 0.10 0.08 0.06 0.04 0.02 NI 0.1 0.2 0.3 0.4 0.5 0.6 Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16
Results b = 0 . 5 , d = 0 . 05 , m 2 = 0 . 01 0.6 K(t) 0.5 I(t) 0.4 NI(t) 0.3 0.2 L(t) 0.1 750. t 150. 300. 450. 600. Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16
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