Apex Consumers and Trophic Downgrading Andrea de Lima Ribeiro (Oc.) - - PowerPoint PPT Presentation

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Apex Consumers and Trophic Downgrading Andrea de Lima Ribeiro (Oc.) - - PowerPoint PPT Presentation

Sep 04, 2023 440 likes •922 views

Apex Consumers and Trophic Downgrading Andrea de Lima Ribeiro (Oc.) Carlos Andrs Marcelo-Servn (Biol.) Carolina Arruda Moreira (Phys.) Cecilia Andreazzi (Biol.) Csar Parra (Phys.) Henrique Rubira (Undergrad.) II Southern-Summer School

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SLIDE 1

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

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SLIDE 2

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

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SLIDE 3

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

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SLIDE 4

The system

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 4 / 16

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Populations

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 5 / 16

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SLIDE 6

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

KEVIN D. LAFFERTY1

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

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SLIDE 7

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

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SLIDE 8

Mathematical Model - Equations

dK dt = K(1 − K) − a(U1 + U2)K dU1 dt = −b U1 U1 + U2 + Uc L − m0U1 − gm1U1 + rU1U2 dU2 dt = ac(U1 + U2)K − b U2 U1 + U2 + Uc L − m0U2 − rU1U2 dL dt = bd U1 + U2 U1 + U2 + Uc L − m2L K, U1, U2, 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 m0, m1, m2: mortality rates of urchins (natural causes), urchins (infection) and lobsters g: multiplying factor Uc: urchin critical density

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 8 / 16

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SLIDE 9

Mathematical Model - Equations

g(U) = 1 + α1 − e−β(U−Uc) 1 + e−β(U−Uc)

0.5 1.0 1.5

g(U)

Uc U

α = 1.0 β = 5.0

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 9 / 16

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SLIDE 10

Results

Sub-system: kelps and not infected urchins dK dt = K(1 − K) − aU2K dU2 dt = acU2K − m0U2 Important fixed points:

1 K∗ = 1, U2∗ = 0: stable for m0 > ac 2 K∗ = m0

ac , U2∗ = ac − m0 a2c : exists and it’s stable for m0 < ac

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 10 / 16

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SLIDE 11

Results

a = 1.0, c = 0.3, m0 = 0.5

14. 28. 42. 56. 70.

t

0.2 0.4 0.6 0.8 1.0 NI (t) K(t) Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 10 / 16

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SLIDE 12

Results

a = 1.0, c = 0.3, m0 = 0.5

0.2 0.4 0.6 0.8 1.0 K 0.1 0.2 0.3 0.4 NI

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 10 / 16

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SLIDE 13

Results

a = 1.0, c = 0.3, m0 = 0.03

20. 40. 60. 80. 100.

t

0.2 0.4 0.6 0.8 1.0 NI(t) K(t) Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 10 / 16

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SLIDE 14

Results

a = 1.0, c = 0.3, m0 = 0.03

0.1 0.2 0.3 0.4 K 0.4 0.6 0.8 1.0 NI

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 10 / 16

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SLIDE 15

Results

Sub-system: kelps, infected and not infected urchins (lobsters have been removed) dK dt = K(1 − K) − a(U1 + U2)K dU1 dt = −m0U1 − gm1U1 + rU1U2 dU2 dt = ac(U1 + U2)K − m0U2 − rU1U2

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16

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SLIDE 16

Results

Important fixed points:

1 K∗ = m0

ac , U1∗ = 0, U2∗ = ac − m0 a2c : stable for r < a2c(m0 + gm1) ac − m0

2

K∗ = 1 2acr [r(ac + m0 + gm1) −

  • r (r(ac + m0 + gm1)2 + 4ac(m0 + gm1)(−r + agm1))
  • U1

∗ =

1 2a2cr

  • r(ac − m0 − gm1) − 2a2c(m0 + gm1)

+

  • r (r(ac + m0 + gm1)2 + 4ac(m0 + gm1)(−r + agm1))
  • U2

∗ = m0 + gm1

r exists and it’s stable for r > a2c(m0 + gm1) ac − m0

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16

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SLIDE 17

Results

r = 0.1, α = 1.0, m1 = 0.05, Uc = 0.5

20. 40. 60. 80.

  • 100. t

0.2 0.4 0.6 0.8 1.0 NI(t) I(t) K(t)

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16

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SLIDE 18

Results

r = 0.1, α = 1.0, m1 = 0.05, Uc = 0.5

0.05 0.10 0.15 K 0.02 0.04 0.06 0.08 0.10 I

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16

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Results

r = 0.1, α = 1.0, m1 = 0.05, Uc = 0.5

0.05 0.10 0.15 K 0.75 0.80 0.85 0.90 0.95 1.00 NI

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16

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SLIDE 20

Results

r = 0.1, α = 1.0, m1 = 0.05, Uc = 0.5

0.02 0.04 0.06 0.08 0.10 I 0.75 0.80 0.85 0.90 0.95 1.00 NI

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16

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SLIDE 21

Results

r = 0.5, α = 1.0, m1 = 0.05, Uc = 0.5

20. 40. 60. 80.

  • 100. t

0.2 0.4 0.6 0.8 1.0 NI(t) I(t) K(t)

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16

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Results

r = 0.5, α = 1.0, m1 = 0.05, Uc = 0.5

0.05 0.10 0.15 0.20 0.25 0.30 K 0.1 0.2 0.3 0.4 0.5 I

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16

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SLIDE 23

Results

r = 0.5, α = 1.0, m1 = 0.05, Uc = 0.5

0.05 0.10 0.15 0.20 0.25 0.30 K 0.2 0.4 0.6 0.8 1.0 NI

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16

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SLIDE 24

Results

r = 0.5, α = 1.0, m1 = 0.05, Uc = 0.5

0 1 0 2 0 3 0 4 0 5 I 0.2 0.4 0.6 0.8 1.0 NI

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16

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SLIDE 25

Results

Effect of increased mortality: r = 0.5, α = 5.0, m1 = 0.05, Uc = 0.5

20 40 60 80 100

t

0.2 0.4 0.6 0.8 1.0 NI(t) I(t) K(t)

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 11 / 16

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Results

All four populations (reintroduction of lobsters) dK dt = K(1 − K) − a(U1 + U2)K dU1 dt = −b U1 U1 + U2 + Uc L − m0U1 − gm1U1 + rU1U2 dU2 dt = ac(U1 + U2)K − b U2 U1 + U2 + Uc L − m0U2 − rU1U2 dL dt = bd U1 + U2 U1 + U2 + Uc L − m2L

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16

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Results

Important fixed points:

1 K∗ = 1 2acr [r(ac + m0 + gm1) −

  • −2acr2(m0 + gm1) + r2(m0 + gm1)2 + a2cr(cr + 4gm1(m0 + gm1))
  • U1∗ =

1 2a2cr

  • acr − 2a2c(m0 + gm1) − r(m0 + gm1)

+

  • −2acr2(m0 + gm1) + r2(m0 + gm1)2 + a2 (4crgm0m1 + cr (cr + 4g2m12))
  • U2∗ = m0 + gm1

r L∗ = 0

stable for b < b0

b0 = m2 d

  • 1 +

a2cUc ac − m0

  • Group 4 (ICTP)

II Southern-Summer School on Mathematical Biology 12 / 16

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SLIDE 28

Results

Important fixed points:

2 K∗ = 1 − am2Uc bd − m2 U1∗ = 0 U2∗ = m2Uc bd − m2 L∗ = 1 (bd − m2)2 dUc

  • (ac − m0)(bd − m2) − a2cm2Uc
  • exists for b > b0 and it’s stable for b0 < b < b1 with stable limit cycle

for b > b1 b0 = m2 d

  • 1 +

a2cUc ac − m0

  • b1: no analytical expression

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16

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Results

b = 0.05, d = 0.05, m2 = 0.01

60 120 180 240 300

t

0.1 0.2 0.3 0.4 0.5 0.6 L(t) NI(t) I(t) K(t)

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16

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Results

b = 0.05, d = 0.05, m2 = 0.01

0.05 0.10 0.15 0.20 0.25 0.30 K 0.1 0.2 0.3 0.4 0.5 0.6 NI

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16

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SLIDE 31

Results

b = 0.05, d = 0.05, m2 = 0.01

0.1 0.2 0.3 0.4 0.5 I 0.1 0.2 0.3 0.4 0.5 0.6 NI

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16

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Results

b = 0.05, d = 0.05, m2 = 0.01

0.1 0.2 0.3 0.4 0.5 I 0.02 0.04 0.06 0.08 0.10 L

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16

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SLIDE 33

Results

b = 0.05, d = 0.05, m2 = 0.01

0.1 0.2 0.3 0.4 0.5 0.6 NI 0.02 0.04 0.06 0.08 0.10 L

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16

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SLIDE 34

Results

b = 0.5, d = 0.05, m2 = 0.01

150. 300. 450. 600.

  • 750. t

0.1 0.2 0.3 0.4 0.5 0.6 L(t) NI(t) I(t) K(t)

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16

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Results

b = 0.5, d = 0.05, m2 = 0.01

0.1 0.2 0.3 0.4 I 0.10 0.15 0.20 0.25 L

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16

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SLIDE 36

Results

b = 0.8, d = 0.05, m2 = 0.01

100. 200. 300. 400.

  • 500. t

0.2 0.4 0.6 0.8 L(t) NI(t) I(t) K(t)

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16

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Results

b = 0.8, d = 0.05, m2 = 0.01

0.2 0.4 0.6 0.8 K 0.1 0.2 0.3 0.4 0.5 0.6 NI

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16

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SLIDE 38

Results

b = 0.8, d = 0.05, m2 = 0.01

0.2 0.4 0.6 0.8 K 0.10 0.15 0.20 L

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16

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SLIDE 39

Results

b = 0.8, d = 0.05, m2 = 0.01

0.05 0.10 0.15 0.20 0.25 0.30 0.35 I 0.10 0.15 0.20 L

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16

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Results

b = 0.8, d = 0.05, m2 = 0.01

0.1 0.2 0.3 0.4 0.5 0.6 NI 0.10 0.15 0.20 L

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16

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SLIDE 41

Results

Different initial conditions

0.1 0.2 0.3 0.4 0.5 0.6 NI 0.05 0.10 0.15 0.20 L

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16

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Results

Bifurcation and onset of limit cycle

b0 b1

b

Lmin Lmax

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 12 / 16

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SLIDE 43

Magnitude of top-down control by pathogens versus predators

a 0.5 1.0 1.5 i 0.5 1.0 1.5 K

  • 0.04
  • 0.01

0.02 0.05 0.08 0.11 0.14 0.16 0.19 0.22

Difference between final Kelp population with and without infection

b 0.5 1.0 1.5 a 0.5 1.0 1.5 K 0.00 0.11 0.22 0.33 0.44 0.55 0.66 0.77 0.88 0.99

Difference between final Kelp population with and without lobsters

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 13 / 16

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Conclusions

  • The mathematical model describes well the dynamics of the

system;

  • The conditions required for the top-down control of the system by

lobsters were determined;

  • The epidemics exerts a top-down control on the urchin population,

however the magnitude of this control is smaller than the control exerced by the apex predator.

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 14 / 16

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SLIDE 45

Future projections of the work

  • Evaluate fishing effort effect on the dynamics of the system.

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 15 / 16

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SLIDE 46

References

Nelson G. Hairston; Frederick E. Smith; Lawrence B. Slobodkin., Community Structure, Population Control, and Competition. The American Naturalist, Vol. 94, No. 879. (Nov. - Dec., 1960), pp. 421-425. Borer & Graner. Top-down and bottom-up regulation of communities., The Princeton guide do Ecology, ch 06.III, 2009. R.T. Paine. Food Webs: Linkage, Interaction Strenght and Community Infrastructure. The Journal of Animal Ecology, Vol. 49, No. 3.(Oct., 1980), pp. 666-685.

Group 4 (ICTP) II Southern-Summer School on Mathematical Biology 16 / 16