Post on 21-Sep-2020
Control of a model of competition
between two animal species
Gianmario Alfonso Varchetta
30/11/2018
- Italian student of automatic engineering
- Goal: Final Thesis work
- Erasmus plus student
- Erasmus period: 3 months (November-January)
Control Strategy
- Control of the system using
feedback control:
π₯ = π π₯, π’π¦ = β(π₯)
π₯ π‘ = π π‘ππ‘π π£ππππππππ π’ = ππππ‘πππ ππππ’π‘
π¦(π₯) = ππ’π‘ππ’π‘
Aim: regulate the system output at a fixed value
The Modelππ₯1ππ‘
= π₯1 π β π₯1 β ππ₯2
ππ₯2ππ‘
= π₯2(π β π₯2 β π₯1)
ππ(t) = population of
species1 (i.e. rabbits)
ππ t = population of
species 2 (i.e. Sheep)
π₯, π¦ β₯ 0
β’ Competition for the
same food supply
and the amount
available is limited.
β’ Each species would
grow to its carrying
capacity in the
absence of the other.
β’ When both species
coexist they start fighting for food.
Equilibriaβ’ π₯π΄ = (0; 0)β’ π₯π΅ = 0; 2β’ π₯πΆ = 3; 0β’ π₯π· = (1; 1)
π½ =3 β 2π₯1 β 2π₯2 β2π₯1
βπ₯2 2 β π₯1 β 2π₯2
π₯π΄ ππ π ππππππππ ππππ.π₯π΅ ππ π π π‘ππππ ππππ.
π₯πΆ ππ π π π‘ππππ ππππ.π₯π· ππ π π πππππ.
β π = 3 π = 2 π = 2
Phase portraitβ’ Principle of
competitive
exclusion: two
species
competing for the
same limited food
typically cannot
coexistβ’ Basins and their
boundaries
partition the phase
plane into regions of different long-term behavior
Dependence on:
Food availabilitySpecies rates
How to control
o Control on food
o Control on species rateso culling and fertility control
- Proportional error controller:
π’ = πΎ1 π₯1πππ β π₯1 + πΎ2 π₯2πππ β π₯2
First results
Varying πΎ1 and πΎ2, the dynamics of the closed loop system change
Aim: to stabilize the saddle in (1;1)
Future steps
β’ Study the Jacobian Matrix and its eigenvalues
depending on K
β’ Try to get bigger or smaller the basins of attraction of the
stable nodes
β’ Implement different kind of control and compare the results(i.e. Adding an integral part of error)