Sustainability in a Discrete Predator–Prey Model with Allee Effect

This post is a blog-friendly explanation of my (currently unpublished) research work on a discrete predator–prey model that incorporates:

• Herd behaviour in prey
• Mate-finding Allee effect
• Predator harvesting
• Environmental sustainability

The goal of the work is to mathematically understand how ecosystems can remain stable under resource extraction — without driving any species to extinction.

Why This Problem Matters

Real ecosystems are rarely smooth or continuous.
Populations are measured:

• Daily
• Monthly
• Seasonally

That makes discrete-time models far more realistic than continuous ones.

At the same time:

• Many species exhibit herd behaviour
• Many species suffer from Allee effects (low population → low reproduction)
• Humans introduce harvesting pressure

All three together create fragile, nonlinear dynamics — exactly what this study tries to capture.

Core Ingredients of the Model

Our model includes three key biological mechanisms:

1. Herd Behaviour (Square-Root Functional Response)

Instead of assuming predators interact with all prey uniformly, we assume:

Predators primarily interact with prey on the boundary of a herd.

This leads to a square-root type interaction term, which is biologically more realistic for schooling fish, flocking birds, etc.

2. Mate-Finding Allee Effect

At very low population sizes, individuals:

• Struggle to find mates
• Experience lower reproduction rates

This introduces positive density dependence at low population sizes, which heavily affects stability.

3. Predator Harvesting

We assume proportional harvesting of the predator population, controlled by:

• Harvesting effort E
• Catchability coefficient q

This lets us mathematically study:

How much harvesting an ecosystem can tolerate without collapsing.

From Continuous to Discrete Model

We begin with a continuous predator–prey system involving:

• Logistic prey growth
• Square-root predation
• Predator mortality + harvesting

Then we:

1. Non-dimensionalize the model
2. Remove the square-root singularity using variable transformations
3. Apply the Forward Euler discretization

This yields a fully discrete predator–prey system:

[ s_{n+1} = s_n + hF(s_n, p_n) ] [ p_{n+1} = p_n + hG(s_n, p_n) ]

This is the system we analyze for:

• Fixed points
• Stability
• Bifurcation behaviour
• Long-term sustainability

Existence of Equilibrium Points

The discrete system admits three biologically meaningful equilibria:

1. Trivial equilibrium:
   [ E_0 = (0,0) ] Represents total extinction.

2. Axial equilibrium:
   [ E_1 = (1,0) ] Prey survives, predator goes extinct.

3. Interior (coexistence) equilibrium:
   [ E_2 = (s^, p^) ] Both populations survive together.

The interior equilibrium exists only when:

[ c > (\hat{\alpha} + e)(b+1) ]

This condition mathematically defines when coexistence is even possible.

Stability Analysis (What Happens Over Time?)

Using Jacobian matrix analysis and discrete eigenvalue theory, we classify each equilibrium as:

• Sink (stable)
• Source (unstable)
• Saddle
• Non-hyperbolic

Key Findings:

• The extinction point is always non-hyperbolic
• The prey-only equilibrium can be:
  • Stable
  • Unstable
  • Saddle
    depending on step size h and harvesting
• The coexistence equilibrium experiences:
  • Stability switches
  • Loss of stability via discrete bifurcations
  • Transition into oscillatory population behaviour

This tells us:

Stability is highly sensitive to both biological parameters and harvesting effort.

Bifurcation Behaviour

As harvesting effort and biological parameters change:

• The system transitions from:
  • Stable coexistence
  • To oscillations
  • To complete collapse

These transitions are classic discrete-time bifurcations, marking critical ecosystem thresholds.

This is extremely important for:

• Fisheries
• Wildlife management
• Renewable resource economics

Numerical Simulation

By selecting biologically valid parameter values, we numerically demonstrate:

• Shifts in equilibrium locations
• Changes in nullcline intersections
• Transition between stable and unstable regimes

Graphically, increasing conversion efficiency c:

• Strengthens predator survival
• Shifts coexistence points upward
• Alters stability domains

Main Conclusion

This study shows that:

• Herd behaviour + Allee effect fundamentally change ecosystem stability
• Predator harvesting introduces critical sustainability thresholds
• Discrete models reveal phenomena invisible to continuous models
• There exists a safe harvesting window where:
  • Both species persist
  • Oscillations remain bounded
  • Extinction is avoided

In short:

Sustainability is not just biological — it is dynamically constrained by nonlinear mathematics.

Why This Work Matters to Me Personally

Working on this paper taught me:

• How fragile real ecosystems are under nonlinear effects
• Why harvesting policies must be parameter-aware
• How discrete systems behave very differently from continuous ones
• How mathematics can directly inform environmental decision-making

It strongly shaped my long-term interest in:

• Complex systems
• Applied dynamical systems
• Computational ecology
• Sustainability-driven modeling

Current Status of the Paper

This work is currently unpublished and under academic circulation for future submission.

Once accepted, I will update this post with:

• Official citation
• Journal/conference link
• DOI

Acknowledgement

This work has been carried out in collaboration with:

• Shilpi Pal
• Debopriya Dey
• Puja Supakar
• Ishita Majumdar

The research was conducted with the academic support of:

Narula Institute of Technology, Agarpara, Kolkata
Departments of Basic Science & Humanities and Computer Science & Engineering

I am sincerely grateful to the faculty and collaborators for their guidance, support, and encouragement throughout the development of this work.

TL;DR

• We built a discrete predator–prey model
• Included herd behaviour + Allee effect + harvesting
• Found strict mathematical conditions for coexistence
• Identified safe harvesting thresholds
• Showed how nonlinear dynamics control sustainability