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The Effect of Predator Growth Rate in a Prey-Predator

System Incorporating Intra-Specific Competition Using

Bifurcation Analysis

NAME: HASUAN AMIN BIN MOHD ROSDI

SUPERVISOR : MADAM ZATI IWANI BINTI ABDUL MANAF

FACULTY OF SCIENCE COMPUTER AND MATHEMATICS

1) ABSTRACT 2) PROBLEM STATEMENT

This research studies how the predator's growth rate affects the balance between predators and Most existing models focus on harvesting or toxins but ignore predator’s natural growth

prey. A mathematical model was created to include both predator growth and competition rate and competition

among predators. Using tools like MATLAB and XPPAUT, the study found that when the This research fills the gap by studying how predator growth affects survival of both

predator grows too slowly, it goes extinct, and prey populations increase uncontrollably. When species.

the predator grows at a higher rate, both species can survive together. The study helps

understand how to manage ecosystems and maintain balance between species.

3) OBJECTIVES 4) METHODOLOGY & IMPLEMENTATION

1) To formulate a prey-predator model incorporating growth rate of predator and intraspecific PHASE 1: To formulate a prey-predator model CORRESPONDING JACOBIAN MATRIX :

competition between predators. incorporating growth rate of predator and predator

2) To perform stability analysis and determine equilibrium conditions for population survival or competition

extinction.

3) To analyze prey-predator dynamics by varying the parameters of growth rate of predator using Original model:

bifurcation analysis

5) RESULTS AND DISCUSSION

STABILITY ANALYSIS :

1) Summary of stability and bifurcation 3) 3-D phase plane for the system

analysis by varying the growth rate of 1.The equilibrium point

New model:
predator, e

PHASE 2: To perform stability analysis and determine

equilibrium conditions for population survival or

extinction.

Determine the equilibrium point:
2. The equilibrium point

Let

(a): Phase plane for growth rate of predator, e = 0.15
Case 1: e = 0.15

Predators grow slowly.

Predators cannot survive (they go extinct).
Calculate Jacobian Matrix:
Only prey remains stable and continues to grow.

The ecosystem becomes unbalanced, dominated by Let

prey.

Case 1: e = 0.4 Then,

Predators grow faster

Both predators and prey coexist in a stable balance.

The system reaches a stable equilibrium, where

neither species goes extinct.

The ecosystem stays balanced
(b): Phase plane for growth rate of predator, e = 0.4 Calculate eigenvalue by letting :

3. The equilibrium point

2) Slicing the one parameter bifurcation 4) Time series graph for different value of e

diagram at e = 0.15 and e = 0.4 for x and y

Critical points:
populations

PHASE 3: To analyze prey-predator dynamics by

varying the parameters of growth rate of predator

using bifurcation analysis

(a) Time series for growth rate of predator,

e = 0.15

Stability analysis :
One parameter bifurcation analysis:
The label B shows the transcritical bifurcation
XPPAUT and Auto software to generate bifurcation
point where predator growth rate e = 2.62. diagram.

At e = 0.15, E (0.5882, 0): Stable (prey survives, MATLAB software to acquire steady-state diagrams
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predator extinct). to analyze stable states by plotting graph phase

E (0.6583, -0.0596): Unstable (not a real plane and time series.
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coexistence since predator value is negative).

At e = 0.4 , E (0.5882, 0): Becomes unstable 7) RECOMMENDATION
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(saddle node). For future research, it is recommended to explore how predator growth rate interacts with other factors such as prey availability, food

E (0.502, 0.074): Becomes stable (both predator (b) Time series for growth rate of predator, quality, habitat conditions, and seasonal changes. Adding more ecological elements to the model, like resource limitations or
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competition for food, can make predictions more realistic. Researchers could also consider including different stages of predator life
and prey survive together). e = 0.4 cycles or study how predator growth varies across different environments. These improvements will help create a deeper understanding

of predator-prey interactions and support better strategies for conserving ecosystems and managing wildlife populations effectively

6) CONCLUSION

The research successfully built a predator-prey model A transcritical bifurcation point was found, showing a When predator growth is high enough, both predator and

focusing on predator growth rate and competition among shift in system stability when the predator’s growth rate prey can coexist in balance

predators. changes. The study achieved all objectives and highlights the

Stability analysis showed how changes in predator When predator growth is too low, predators may go importance of managing predator growth rates to maintain

growth affect population survival or extinction. extinct, and prey populations grow uncontrollably ecosystem stability