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The lotka-volterra model, a mathematical representation of predator-prey relationships. The model examines the interactions between prey (h) and predators (p), their population growth rates, and the stability of the system. Tanner (1975) discussed the model's stability, focusing on the critical point where predator and prey isoclines cross, which can result in stable or unstable equilibria. The document also discusses the impact of predator resource limitations and prey self-limitation on the system.
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H = number of prey
p^
y
r = prey population growth rateb = attack rate
P = number of predators
c = predator population growthrate due to predationk = rate of predator decline in
p
absence of prey
dH / dt < 0dP / dt < 0
dH / dt < 0dP / dt > 0
P
r/b P
k/c
dH / dt > 0dP / dt > 0
dH / dt > 0dP / dt < 0
H
-^
-^
-^
Stable focus whenthe critical pointfalls to the right offalls to the right ofthe prey zeroisocline peak for all values of s/r
When the criticalpoint falls to theleft of the prey zeroleft of the prey zeroisocline peak, 2)
limit cycle
if s/r
small
When the criticalpoint falls to theleft of the prey zeroleft of the prey zeroisocline peak, 3) unstable focus
if
s/r small and K isvery large ā
y^
g
extinction; nocoexistence
Once again, sincethe critical pointfalls to the right offalls to the right ofthe prey zeroisocline peak, a stable focus resultsfor all values of s/r
Again, since thecritical point falls tothe right of thethe right of theprey zero isoclinepeak, a stable results for all valuesof s/r
The preypopulation can get
Unstable focus
āstuckā at very lowdensity unlesspredation rates
Stable focus
predation rates drop substantially ,called a predatorpitpit
Stable focus āP
d t
Pitā
āP
redator Pitā
sparrow hawk / house sparrow and
-^
Mink / muskratMink
/ muskrat
-^
Lynx / snowshoe hare
-^
-^
-^
-^
-^
