Very brief mathematical introduction to the population dynamics of insects. Last part, on spatial spread is new. Joint work with W.A.C. Godoy and R.M. Coutinho.

1. ift-unesp
population dynamics of insects
Roberto A. Kraenkel
Institute for Theoretical Physics - UNESP
São Paulo, BR
kraenkel@ift.unesp.br

2. ift-unesp
outline

3. ift-unesp
outline
• crash course on population dynamics

4. ift-unesp
outline
• crash course on population dynamics
• what is special with insects

5. ift-unesp
outline
• crash course on population dynamics
• what is special with insects
• competition & predation

6. ift-unesp
outline
• crash course on population dynamics
• what is special with insects
• competition & predation
• insects in space & time

7. crash course on
ift-unesp
population dynamics

8. crash course on
ift-unesp
population dynamics
• it’s about populations, not individuals

9. crash course on
ift-unesp
population dynamics
• it’s about populations, not individuals
• mathematically, a population is described
either by its density or by the total number
of individuals in a region

10. crash course on
ift-unesp
population dynamics
• it’s about populations, not individuals
• mathematically, a population is described
either by its density or by the total number
of individuals in a region
• to describe its dynamics in space and time
we have to model the main processes the
population is subject to

11. ift-unesp
processes

12. ift-unesp
processes
• Growth

13. ift-unesp
processes
• Growth
• by reproduction

14. ift-unesp
processes
• Growth
• by reproduction
• by consumption of abiotic resources

15. ift-unesp
processes
• Growth
• by reproduction
• by consumption of abiotic resources
biotic

16. ift-unesp
processes
• Growth
• by reproduction
• by consumption of abiotic resources
biotic abiotic

17. ift-unesp
saturation

18. ift-unesp
saturation
• Growth has to saturate:

19. ift-unesp
saturation
• Growth has to saturate:
logistic type

20. ift-unesp
interactions i

21. ift-unesp
interactions i
• competition for resources : 2-species,
Lotka-Volterra type.

22. ift-unesp
interactions i
• competition for resources : 2-species,
Lotka-Volterra type.
• principle of competitive exclusion: if strong
enough, competition leads to exclusion of
one species.

23. ift-unesp
interactions i
• competition for resources : 2-species,
Lotka-Volterra type.
• principle of competitive exclusion: if strong
enough, competition leads to exclusion of
one species.

24. ift-unesp
interactions i
• competition for resources : 2-species,
Lotka-Volterra type.
• principle of competitive exclusion: if strong
enough, competition leads to exclusion of
one species.

25. ift-unesp
interactions ii

26. ift-unesp
interactions ii
• predation ( trophic interactions):

27. ift-unesp
interactions ii
• predation ( trophic interactions):
• asymmetric -- one predator, one prey

28. ift-unesp
interactions ii
• predation ( trophic interactions):
• asymmetric -- one predator, one prey
• Lotka-Volterra

29. = P (cV − d) ift-unesp
dt
interactions ii
dV
= V (a − bP )
• predation ( trophic interactions):
dt
• asymmetric -- one predator, one prey
• Lotka-Volterra
dP
= P (cV − d)
dt

30. ift-unesp
interactions ii
• predation ( trophic interactions): C
y
• asymmetric -- one predator, one prey c
• Lotka-Volterra l
e
s

31. ift-unesp
movement

32. ift-unesp
movement
• macroscopically, the most simple
assumption is that of a diffusive spreading
of the population.

33. ift-unesp
movement
• macroscopically, the most simple
assumption is that of a diffusive spreading
of the population.

34. ift-unesp
movement
• macroscopically, the most simple
assumption is that of a diffusive spreading
of the population.
This is compatible with a brownian movement of
individuals

35. ift-unesp
movement
• macroscopically, the most simple
assumption is that of a diffusive spreading
of the population.
This is compatible with a brownian movement of
individuals
If you put diffusion + growth + saturation:

36. ift-unesp
movement
• macroscopically, the most simple
assumption is that of a diffusive spreading
of the population.
This is compatible with a brownian movement of
individuals
If you put diffusion + growth + saturation:

37. ift-unesp
insects

38. ift-unesp
insects

39. ift-unesp
insects
are a class within
the arthropods
that have an
exoskeleton, a
three-part body
(head, thorax, and
abdomen), three
pairs of jointed
legs, compound
eyes, and two
antennae.

40. ift-unesp
insects
are a class within
the arthropods
that have an
exoskeleton, a
three-part body
(head, thorax, and
abdomen), three
pairs of jointed
Most insects put eggs, which hatch to give legs, compound
birth to larvae eyes, and two
antennae.
Larvae undergo metamorphosis: after a
pupae or nymphae stage, they become adults

41. population biology of
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insects

42. population biology of
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insects
• What’s special with insects?

43. population biology of
ift-unesp
insects
• What’s special with insects?
• From the point of view of the population dynamics:

44. population biology of
ift-unesp
insects
• What’s special with insects?
• From the point of view of the population dynamics:
• the ecological function of larvae and adult stage are
different.

45. population biology of
ift-unesp
insects
• What’s special with insects?
• From the point of view of the population dynamics:
• the ecological function of larvae and adult stage are
different.
• usually larvae are responsible for the populational
regulation

46. population biology of
ift-unesp
insects
• What’s special with insects?
• From the point of view of the population dynamics:
• the ecological function of larvae and adult stage are
different.
• usually larvae are responsible for the populational
regulation
• adults disperse

47. population biology of
ift-unesp
insects
• What’s special with insects?
• From the point of view of the population dynamics:
• the ecological function of larvae and adult stage are
different.
• usually larvae are responsible for the populational
regulation
• adults disperse
• adults ==> larvae==> adults ==> ....

48. population biology of
ift-unesp
insects
• What’s special with insects?
• From the point of view of the population dynamics:
• the ecological function of larvae and adult stage are
different.
• usually larvae are responsible for the populational
regulation
• adults disperse
• adults ==> larvae==> adults ==> ....
• dynamics can be discrete in time: non overlapping
generations

49. population dynamics of
ift-unesp
insects

50. population dynamics of
ift-unesp
insects
• The simplest model is due to Prout &
McChesnay (1985)

51. population dynamics of
ift-unesp
insects
• The simplest model is due to Prout &
McChesnay (1985)
• It is a discrete time model

52. population dynamics of
ift-unesp
insects
• The simplest model is due to Prout &
McChesnay (1985)
• It is a discrete time model
• Adults (v) generate larvae (u)

53. population dynamics of
ift-unesp
insects
• The simplest model is due to Prout &
McChesnay (1985)
• It is a discrete time model
• Adults (v) generate larvae (u)
• Larvae generate the next generation of
adults

54. population dynamics of
ift-unesp
insects
• The simplest model is due to Prout &
McChesnay (1985)
• It is a discrete time model
• Adults (v) generate larvae (u)
• Larvae generate the next generation of
adults
• nonlinear terms are such as not to
generate negative populations

55. population dynamics of
ift-unesp
insects
ut = Svt exp(−svt )
1
vt+1 = F ut exp(−f vt )
2

56. population dynamics of
ift-unesp
insects
ut = Svt exp(−svt )
1
vt+1 = F Svt exp(−(f + s)vt )
2

57. population dynamics of
ift-unesp
insects
Let us now look at some
examples involving a particular
species : blowflies of the
species Chrysomya albiceps

58. population dynamics of
ift-unesp
insects

59. ift-unesp
Chrysomya albiceps

60. ift-unesp
Chrysomya albiceps
• Facts:

61. ift-unesp
Chrysomya albiceps
• Facts:
• originally from Africa

62. ift-unesp
Chrysomya albiceps
• Facts:
• originally from Africa
• introduced in the Americas circa 1975

63. ift-unesp
Chrysomya albiceps
• Facts:
• originally from Africa
• introduced in the Americas circa 1975
• it dislocated native blowflies ( Cochliomya
macellaria)

64. ift-unesp
Chrysomya albiceps
• Facts:
• originally from Africa
• introduced in the Americas circa 1975
• it dislocated native blowflies ( Cochliomya
macellaria)
• it predates other blowflies

65. ift-unesp
Chrysomya albiceps
• Facts:
• originally from Africa
• introduced in the Americas circa 1975
• it dislocated native blowflies ( Cochliomya
macellaria)
• it predates other blowflies
• its introdiction occured tpgether with the
introduction of one of its prey , C.
megachephala.

66. competition &
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predation
with Gabriel A. Maciel

67. competition &
ift-unesp
predation
• Two species model
with Gabriel A. Maciel

68. competition &
ift-unesp
predation
• Two species model
• Competition
with Gabriel A. Maciel

69. competition &
ift-unesp
predation
• Two species model
• Competition
• Predation ( Intraguild predation)
with Gabriel A. Maciel

70. competition &
ift-unesp
predation
• Two species model
• Competition
• Predation ( Intraguild predation)
• Each species has two stages
with Gabriel A. Maciel

71. competition &
ift-unesp
predation
• Two species model
• Competition
• Predation ( Intraguild predation)
• Each species has two stages
Competition and predation
only in larval stage with Gabriel A. Maciel

72. competition &
ift-unesp
predation
• Two species model
• Competition
• Predation ( Intraguild predation)
• Each species has two stages
Competition and predation
only in larval stage with Gabriel A. Maciel

73. competition &
ift-unesp
predation
with Gabriel A. Maciel

74. ift-unesp
invasion
with Renato M. Coutinho

75. ift-unesp
invasion
• Model for the spatial distribution of C.
albiceps
with Renato M. Coutinho

76. ift-unesp
invasion
• Model for the spatial distribution of C.
albiceps
• discrete in time
with Renato M. Coutinho

77. ift-unesp
invasion
• Model for the spatial distribution of C.
albiceps
• discrete in time
• continous in space.
with Renato M. Coutinho

78. ift-unesp
invasion
• Model for the spatial distribution of C.
albiceps
• discrete in time
• continous in space.
only adults disperse single species model
with Renato M. Coutinho

79. ift-unesp
invasion
uses a gaussian kernel
• Model for the spatial distribution of C.
albiceps
• discrete in time
• continous in space.
only adults disperse single species model
with Renato M. Coutinho

80. ift-unesp
invasion
generalizes M. Kot resuts uses a gaussian kernel
• Model for the spatial distribution of C.
albiceps
• discrete in time
• continous in space.
only adults disperse single species model
with Renato M. Coutinho

81. ift-unesp
invasion
with Renato M. Coutinho

82. ift-unesp
invasion
with Renato M. Coutinho

83. ift-unesp
invasion
propagation front with
with Renato M. Coutinho constant speed

84. with Renato M. Coutinho ift-unesp
invasion

85. with Renato M. Coutinho ift-unesp
invasion
• Compare with observations of C. albiceps in
Brazil?

86. with Renato M. Coutinho ift-unesp
invasion
• Compare with observations of C. albiceps in
Brazil?
• Need data on dispersion + lab data on vital
rates

87. with Renato M. Coutinho ift-unesp
invasion
• Compare with observations of C. albiceps in
Brazil?
• Need data on dispersion + lab data on vital
rates
• Dispersion data available for the same
species in South Africa ( 1984)

88. with Renato M. Coutinho ift-unesp
invasion
• Compare with observations of C. albiceps in
Brazil?
• Need data on dispersion + lab data on vital
rates
• Dispersion data available for the same
species in South Africa ( 1984)
• Re-analisys of SA data + lab mesurements

89. with Renato M. Coutinho ift-unesp
invasion
• Compare with observations of C. albiceps in
Brazil?
• Need data on dispersion + lab data on vital
rates
• Dispersion data available for the same
species in South Africa ( 1984)
• Re-analisys of SA data + lab mesurements

90. with Renato M. Coutinho ift-unesp
invasion
• Compare with observations of C. albiceps in
Brazil?
• Need data on dispersion + lab data on vital
rates
• Dispersion data available for the same
species in South Africa ( 1984)
• Re-analisys of SA data + lab mesurements

91. with Renato M. Coutinho ift-unesp
invasion
Prediction for invasion speed is between 0.3 to 2. 2 km per
day

92. with Renato M. Coutinho ift-unesp
invasion
Prediction for invasion speed is between 0.3 to 2. 2 km per
day
which corresponds to historical records of the invasion

93. with Renato M. Coutinho ift-unesp
invasion
Prediction for invasion speed is between 0.3 to 2. 2 km per
day
which corresponds to historical records of the invasion
Nice!!

94. ift-unesp
final comments

95. ift-unesp
final comments
• Population dynamics of insects goes trough
modelling different stages

96. ift-unesp
final comments
• Population dynamics of insects goes trough
modelling different stages
• Each stage may have different ecological
functions

97. ift-unesp
final comments
• Population dynamics of insects goes trough
modelling different stages
• Each stage may have different ecological
functions
• Data are rare and not very precise

98. Thank you for your
ift-unesp
attention
• kraenkel@ift.unesp.br