How can Lotka-Volterra equations and Verhulst’s logistic growth models be applied and compared to model the population and behavior of European rabbits (Oryctolagus cuniculus) in the UK from 1995 to 2024?SL
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A
Official IB Result
27/34
Criteria A: Focus and Method
4/6
0
3
6
Criteria StrandsPro
A.1Topic communication and explanation
Excellent
A.2Research question clarity and focus
Good
A.3Research methodology
Moderate
Criteria Feedback
- The student states a clear mathematical topic and connects it to a specific ecological case with two named models.
- The student's investigation keeps a recognizable focus on comparing the two modelling approaches throughout the report.
- The student shows a purposeful method by deriving and applying models rather than only describing them.
- The student's comparison question is still broader than it needs to be, so the basis for judging the models is not fully sharp.
- The opening includes some general ecological framing that delays the mathematical purpose of the investigation.
- Some parts of the method would benefit from a more explicit plan for how the models will be compared using the same evidence.
1.1·Weakness
Page 1
The research question is explicit, but the comparison criterion is still too broad. The student should state in advance how “compared” will be judged — for example by predictive error, fit to observed data, or explanatory power — so the investigation has a sharper method and a clearer conclusion.
1.2·Strength
Page 1
The topic is communicated clearly and mathematically appropriately: the student identifies two specific modelling approaches, the species, location, and time span. This gives the investigation a precise ecological and mathematical frame from the outset.
1.3·Strength
Page 3
The introduction successfully sets up the investigation’s purpose and scope. Naming both models here helps the reader understand the direction of the essay and why the subsequent derivations are relevant.
1.4·Suggestion
Page 3
The student could tighten the opening by reducing the broad ecological framing and moving sooner to the rabbit/fox case study. A more focused introduction would make the mathematical purpose more immediate and avoid diluting the central investigation.
1.5·Weakness
Page 4
This background is useful, but it stays descriptive. The student should connect the model more directly to the exact variables and assumptions that will matter later in the rabbit investigation, otherwise the methodology reads like general notes rather than a purposeful plan.
1.6·Strength
Page 7
The student shows appropriate methodological awareness by introducing the Lotka–Volterra framework and linking it to a biological predator–prey setting. This helps justify why the second model is relevant for the comparison.
1.7·Weakness
Page 8
The assumptions are listed, but the student does not yet explain which of them are acceptable for the rabbit–fox context and which may distort the model. A stronger methodology would evaluate the suitability of each assumption before using it in later parameter choices.
1.8·Question
Page 14
What justifies choosing the minimum territorial range rather than the midpoint or an empirically averaged value? Reflecting on that choice would help the student show informed parameter selection rather than selecting the value that gives the largest carrying capacity.
Criteria B: Knowledge and Understanding
4/6
0
3
6
Criteria StrandsPro
B.1Subject knowledge application
Moderate
B.2Use of terminology and concepts
Moderate
Criteria Feedback
- The student uses a strong range of mathematical terminology accurately in many places, including modelling, equilibrium, eigenvalues, and Jacobian matrix.
- The student shows real understanding by deriving the logistic solution and using linearization ideas in the predator–prey section.
- The student's mathematics is usually connected to the ecological context rather than presented as isolated formulas.
- Some interpretations are mathematically inaccurate, especially around the shape and long-term behaviour of the logistic model.
- The stability classification needs more care, because the language used for the equilibrium type is not always consistent with the eigenvalues obtained.
- A few conclusions overstate what the calculations actually show.
2.1·Strength
Page 5
The student demonstrates solid mathematical knowledge by deriving the logistic solution step by step rather than simply quoting the formula. This shows understanding of both the calculus technique and its role in the model.
2.2·Strength
Page 13
The student correctly uses eigenvalues as a tool for interpreting system behaviour around equilibrium points. This is strong subject knowledge because it links linear algebra to the ecology context, not just to isolated calculations.
2.3·Weakness
Page 18
This is an incorrect interpretation of the graph. For a logistic model with positive population values, the graph does not have a finite x-intercept in the ordinary sense; the population approaches zero asymptotically. The student should not present extinction as occurring at a specific graph intercept unless the equation truly gives one.
2.4·Weakness
Page 18
This interpretation is mathematically inaccurate. A logistic model does not become a negative linear function, and a positive population does not reach zero in finite time; it approaches its limiting behaviour asymptotically. The student should revise the language to match the model’s actual shape.
2.5·Strength
Page 26
The student shows a clear grasp of how to move from the nonlinear system to local linear analysis by computing partial derivatives at a specified point. That is an appropriate mathematical step in a Lotka–Volterra investigation.
2.6·Weakness
Page 28
The linear stability interpretation is mostly right, but it needs sharper terminology. A complex pair with positive real part indicates an unstable spiral/focus, not a saddle. The student should keep the equilibrium classification consistent with the eigenvalue type being discussed.
2.7·Weakness
Page 29
This conclusion does not follow from the eigenvalue discussion. A saddle requires real eigenvalues of opposite signs, whereas the preceding analysis describes a complex pair with positive real part. The student should separate “unstable spiral/focus” from “saddle” to avoid a classification error.
Criteria C: Critical Thinking
12/12
0
6
12
Criteria StrandsPro
C.1Research quality
Poor
C.2Analysis depth
Poor
C.3Discussion and evaluation
Poor
Criteria Feedback
- The student uses real data and attempts to fit models to observed population values rather than remaining abstract.
- The student makes a genuine effort to evaluate the usefulness of the models and compare their behaviour in an ecological context.
- The student includes analysis steps that go beyond calculation, such as discussing assumptions, equilibrium, and model limitations.
- Several key conclusions rest on a very small evidence base, so the comparisons are not as secure as they could be.
- The parameter work for the predator–prey system is weakened by inconsistent units, which affects the reliability of the later analysis.
- Some claims about accuracy and superiority are stronger than the evidence allows, especially when only limited validation is used.
3.1·Strength
Page 16
The student applies the model to actual data rather than staying abstract, which is a strong start for research quality. Using observed values to solve parameters makes the investigation more purposeful and evidence-based.
3.2·Weakness
Page 19
This analysis is weakened by an incorrect model description. The logistic curve is not a negative linear function, so the limitation should be discussed in terms of the model’s asymptotic shape and simplifying assumptions, not linearity.
3.3·Weakness
Page 20
The conclusion is too strong for a single validation point. One 2016 comparison is not enough to justify broad claims of accuracy; the student should test additional points or clearly frame the result as a limited local check rather than general proof.
3.4·Weakness
Page 23
This sentence is speculative and not analytically useful. The student should replace it with a clearer explanation of how eta and delta are being estimated and why those estimates are reliable or unreliable, rather than claiming they would cause extinction.
3.5·Weakness
Page 24
This calculation mixes percentage changes with derivatives in population units, so the parameter estimate is dimensionally inconsistent. The student should convert all rates into the same units as the differential equation before using them to solve the system.
3.6·Weakness
Page 25
The parameter fitting here is not methodologically secure because it relies on a very small and weakly justified dataset. The student should explain why these particular observations are representative and consider fitting against more than one time point for each species.
3.7·Weakness
Page 31
The comparison is interesting, but it remains partially unsupported. Because the Lotka–Volterra parameters were derived from dimensionally inconsistent rates and a very small dataset, the student should be more cautious about claiming that it “better” describes the ecosystem.
3.8·Strength
Page 31
The student makes a clear evaluative claim that distinguishes the logistic model’s quantitative usefulness from a more detailed ecological model. This is a good sign of analysis because it moves beyond description toward judgement.
Criteria D: Presentation
3/4
0
2
4
Criteria StrandsPro
D.1Structure appropriateness
Good
D.2Layout and formatting
Good
Criteria Feedback
- The student's report has a clear overall structure with an introduction, development, conclusion, and bibliography.
- Figures, equations, and headings are generally used in a way that helps the reader follow the investigation.
- The work is mostly easy to navigate and visually organized.
- Some pages are underfilled or left blank, which makes the layout feel less efficient.
- A few figures would be more useful if their scales and connection to the argument were clearer.
- The final section could be tighter so that the main comparison is easier to absorb quickly.
4.1·Strength
Page 2
The essay has a clear conventional structure, beginning with an introduction and moving into methodology, model application, evaluation, conclusion, and bibliography. That sequence makes the investigation easy to follow.
4.2·Strength
Page 15
The student integrates figures into the argument rather than leaving them decorative. The labelled graph gives the reader a visual reference for the trend being discussed in the text.
4.3·Suggestion
Page 18
The graph would be more informative if the axes were clearly interpretable from the text as well, especially the time scale and population units. A reader should be able to connect the graph directly to the data table without having to infer the scale.
4.4·Weakness
Page 28
The figure is used in the reasoning, but the analysis does not clearly explain how the trajectories in the phase plane support the stated classification. The student should connect the visual evidence to specific features of the system, such as direction of flow near the equilibrium.
4.5·Suggestion
Page 31
The conclusion would be stronger if it were shortened and made more decisive. The student could state the key comparison in one or two sentences and then briefly explain which model was more useful under which assumptions.