Download Beetles

Do They Stay or Do
They Go Now?
Habitat selection of aquatic
beetles and its impact on spatial
distribution
By Drew Hanson & Justin Marleau
May 11th, 2007
PIMS Mathematical Biology
Summer Workshop
Introduction

The most utilized research programme for the
explanation of species distributions and
abundances in community ecology is random
dispersal followed by non-random, site-specific
mortality (Binckley & Resetarits 2005).
 While other research programmes have been
developed involving different mechanisms, such
as neutral theory (see Hubbell 2001), few have
incorporated important behavioural mechanisms
which can possibly alter species distributions
and abundances.
Introduction

One such mechanism is habitat selection, which
involves organisms dispersing and colonizing
patches with the highest expected fitness.
 Our goal is to model this mechanism and
determine if there are significant differences
between the emergent distribution and
abundance of organisms predicted by random
dispersal and non-random site-specific mortality.
The Organism: Beetle


Tropisternus lateralis, a
common predaceous
diving water beetle.
Have an initial dispersal
flight after hatching that
can be over a half-mile
(800 metres) in distance
(Milliger & Schlicht 1968,
Wallace & Anderson
1996).
The Habitat: Ponds

T. lateralis occurs
naturally in small
ponds (like rice
paddies, temporary
woodland ponds) and
is considered to be an
important species in
aquatic systems as a
predator and a prey
(Resetarits 2001).
Proposed Mechanisms of Habitat
Selection

Pond Size: Beetles are attracted to shiny
surfaces, so larger ponds would be more likely
to be seen by beetles flying overhead and
therefore be colonized by beetles after their
dispersal flight (Wallace & Anderson 1996).
 Presence of predator: Certain species of beetles
are hypothesized to be able to detect chemically
the presence of predators once in the pond,
making them more likely to leave ponds with
predators (Binckley & Resetarits 2005).
Other Relevant Biological
Information
 Beetles
in ponds containing predators are
less active than in ponds containing no
predators (Resetarits 2001).
 Ponds can become rapidly saturated with
beetles as they preferentially stay on the
side of ponds, causing the beetles to leave
and colonize other ponds once a threshold
level of abundance is reached (Binckley &
Resetarits 2005).
How does one model the system?

As the beetles are undergoing long-distance
dispersal, we will be using a gravity model
(Bossenbroek et al. 2001).
 A gravity model, which is based on Newton’s law
of gravity, assumes that individuals will be
attracted to large areas, i.e. large ponds in our
model.
 Other components can affect the “attractiveness”
of lakes, such as the nutrient composition and
the presence of predators.
The Breakdown: Outline of Talk
 1.
Creating an appropriate gravity model
and qualitatively analyse it, without regard
to presence of predators.
 2. Introducing the predators, but not
introducing predation, in order to compare
with results of Binckley & Resetarits
(2005).
 3. Introduce predation and multiple
generations for a “realistic” model.
Part I: The Set-Up
Initial Model Assumptions:





All ponds have same concentrations of
nutrients for beetles.
There is no mortality of beetles over the
time-frame considered.
Each pond, depending on its size, has a
carrying capacity of beetles. Beetles will not
leave a pond if it is under carrying capacity.
The model is governed by deterministic
equations.
Part I: The Set-up

Hundred randomly distributed ponds of variable size.
Original population starts at (0,0) at time 0.
Part I: Parameters for initial
deterministic model










T = Total number of beetles in the system of ponds
Tj = Total number of beetles in pond j
Kj = Carrying capacity of pond j
Ai = Balancing coefficient that ensures all beetles leaving pond i
arrive to some pond in the system
cij = distance from pond i to pond j
N0j = Number of beetles arriving from outside the pond system that
arrive to pond j during their dispersal flight
Wj = Attractiveness of pond j
Mi = Number of beetles leaving pond i
a = Distance coefficient
P = Total number of ponds in our system
Part I: The deterministic equations
1
N 0 j  N 0 A0W j c0aj
P
A0  1 /  W j c0aj
j 1
a
j ij
N ij  N i AiW c
2
P
Ai  1 /  W j cija
j 1
M i  Ti  K i , Ti  K i
or
M i  0, Ti  K i
Part I: Parameter values
= area of pond j in m2
 Kj = (1/0.435m2)*60*(area of pond j) (Note:
The (1/0.435m2)*60 term is derived from
Binckley & Resetarits (2005))
 a = 1.9
 T = 15 000
 Wj
Part I Results: 15000 at (0,0) at t=0
Part I: If we start elsewhere?
Part I: If we have more beetles?
Part II: Adding greater realism
 Distance
and size of ponds are not the
only factors determining the abundance
and distribution of beetles, the presence of
fish are also very important.
 Fish act in two ways: they reduce the
attractiveness of ponds by reducing the
perceived carrying capacity of ponds and
they consume beetles.
Part II: Beetle Eating Machine

Enneacanthus obesus: blue-stripped sunfish
Part II: The Effects of the BlueStripped Sunfish
 Can
reduce its prey population by 70% in
a single day at an intermediate size
(3.75g) (Chalcraft & Resetarits 2004).
 Mere presence reduces the attractiveness
of ponds (carrying capacity) by 80% to
beetles and greatly decreases the activity
of beetles remaining in the ponds
(Resetarits 2001, Binckley & Resetarits
2005).
Part II: Investigating non-lethal
impact of predators
 In
order to see if our model could properly
model the non-lethal effects observed by
Resetarits and associates, we re-created
one of their experimental set-ups within
our model framework and we assumed
that the carrying capacity of ponds was the
substantial difference between ponds (as
was the case with Binckley & Resetarits
2005).
Part II: Visual Representation of
Binckley & Resetarits (2005)
Part II : Results
Part III: The “generalized” model

In this section, we allow for predation, multiple
generations of beetles and the carrying capacity
is lowered by the presence of fish.
 As beetles are univoltine (one generation alive
at a time), we assume that beetles lay eggs (10
eggs per adult) near the end of the year, and
expire before the next generation is hatched.
(Zola et al. 1980).
 We also assume that newly-hatched beetles all
leave their original pond to colonize new ponds,
but do not leave the pond system.
Part III: Changes to Equations
Tj – min(0.7*Tj *number of fish in pond
j, 70*number of fish in pond j)
 The above equation takes into account the
voracious appetite of the predators in
modifying the beetle populations.
 Tj =
Part III: Result
Part III: Differences between detecting
and non-detecting beetles
Part III: Differences between detecting
and non-detecting beetles
Part III: Differences between detecting
and non-detecting beetles
 After
one hundred random samplings
(from different starting points), we
discovered:
 Mean of detecting beetles: 5345
 Standard Deviation: 707
 Mean of non-detecting beetles: 4979
 Standard Deviation: 759
Part III: Differences between detecting
and non-detecting beetles
 These
differences are highly dependent on
when the predation, the number of
predators present, the location of the
predators and the birth rates of the
beetles. Greater analysis is needed before
any firm conclusions can be drawn.
Part III: Pros and Cons
PROS
 The beetle population, regardless of initial conditions,
goes to a global steady state over some number of
years.
 The model is straightforward, and seems to accurately fit
limited experimental data that exists.
CONS
 The assumption of habitat selection reduces the overall
capacity of the pond system. This leads to a increased
population density of selective beetles rather than an
increased overall population.
 Birth rates are constant among all beetles, while
realistically, beetles who are living in a pond with
predators have much lower fitness.
Summary

It is possible and useful to model dispersal flights of
beetles as a simple gravity model.
 The gravity model with changes in carrying capacity due
to the presence of predators is capable of generating
results that are similar to those of experimental studies.
 There can be global steady-states independent of initial
conditions and beetle detection strategy if multiple
generations are considered and the birth rates are large
enough.
 The actual dynamics of multiple generations is highly
dependent on parameter values and require greater
study.
Acknowledgements
 We
would like to thank Mark and Caroline
for their insightful advice in the formulation
of our project as well as everyone
participating and organizing this workshop
for creating such a great atmosphere.
References



Binckley, C.A. & Resetarits, W. 2005 Habitat selection
determines abundance, richness and species
composition of beetles in aquatic communities. Biol. Lett.
1, 370-374.
Resetarits, W. 2001 Colonization under threat of
predation: avoidance of fish by an aquatic beetle,
Tropisternus lateralis (Coleoptera: Hydorphilidae).
Oecologia 129, 155-160.
Wallace, J.B. & Anderson, N.H. 1996 Habitat, life history
and behavioral adaptations of aquatic insects. In An
introduction to the aquatic insects of North America (ed.
R.W. Merritt & K.W. Cummins), pp. 41-73. Dubuque,
Kendall/Hunt.
References

Milliger, L.E. & Schlicht, H.E. 1968 Passive
dispersal of viable algae and protozoa by an
aquatic beetle. Trans. Amer. Microsc. Soc. 87,
443-448.
 Hubbell, S.P. 2001 Unified Theory of Biodiversity
and Biogeography. Princeton.
 Bossenbroek, J.M., Clifford, E.K. & Nekola, J.C.
2001 Prediction of long-distance dispersal using
gravity models: zebra mussel invasion of inland
lakes. Ecological Applications, 11, 1778-1788.
References
 Zalom,
F.G., Grigarick, A.A. & Way, M.O.
1980 Habits and relative population
densities of some hydrophilids in California
rice fields. Hydrobiologia, 75, 195-200.
 Chalcraft, D.R. & Resetarits, W.J. 2004
Metabolic rate models and the
substitutability of predator populations.
Journal of Animal Biology, 73, 323-332.